What special considerations must be taken when designing parts for 3D printing?
Designing a part for 3D printing often doesn't seem to have many special considerations, but I have learned the hard way, that there are some things to do differently. This is just a list of things to that one should keep in mind in addition to basic principles of design1 when designing parts, keeping the subsequently slicing the parts in mind too:
Print orientation
There are many ways how you could orient your print, but usually, there is one orientation, that has the least need for support. Look at your part critically and keep this orientation in mind when designing. Especially look at overhangs and bridging, and if you can get away without.
Overhangs
There are 3 sorts of overhangs:

Small overhangs ones that are neglectable.
Long overhangs which can get support.
Overhangs that can't be supported.

When designing parts, you want to make sure you only have type 1 and 2 Overhangs, as type 3 overhangs will sag and fail. Think carefully if you can rotate the piece to get a print orientation that does not need an overhang that can't be supported by an automatically generated support structure. If that is impossible, try to implement a sacrificial piece that can turn the overhang into a bridge.
Smaller overhangs can be made neglectable by adding a phase to the underside. This phase's angle is depending on the printer. In my experience, 70° is something many printers can manage, but I prefer 45° due to the ease of making them. A fuller can work to give a small overhang the needed support, but often has problems for larger overhangs!
Bridges
Overhangs turn into bridges if they are connected on both sides. These either have a limited length, depending on the printer you use or need a support in the center. Check if you really need a bridge or if you can rotate the piece to get away without.
Avoid vertical holes in bridges
It might be something that might surprise but a vertical hole or slot in a bridging part is something that often fails as the bridging strings just sag as they are terminated mid-air without a support structure and finally fall, ruining the bridge. Yet such a support structure sometimes could not be removed in all cases, so something needs to be done differently.
One such a solution is to add a 1-layer sacrificial layer on the bottom of the hole: printing a solid layer by bridging is possible, and the subsequent hole/slot can still be free. It has to be cut free after printing in post-processing though.
round holes in Walls
Round holes in standing walls can become problematic to print once the diameter gets too large. A trick to keep the upper parts of the hole to sag into the cavity, making it undersized or needing to drill it to size later. To prevent this, the upper side of the hole can be adjusted: instead of a round upper rim, turn the hole into a teardrop. This reduces the overhanging area. Keeping a 60° top angle on the hole should be fine.
If the hole is used to key an item to an axle, put the keyway to the top of the print orientation, so it takes the place of the teardrop-tip.
Some more about holes one can learn from Makers Muse (Angus Daveson)
Reduce internal structure
I have seen prints fail for strange reasons. One of them was a piece taken from a straight up industrial design plan, then scaled down. This one resulted in too much tiny internal geometry, resulting in a lot of material and time wasted on printing these internal pieces that nobody could reach, that were fused together for the original gaps were already 0.2mm and less and besides that, there was the occasional print failure.
Removing any non-essential internal geometry lessens not only the printer's load, but speeds up the print, lessens the material waste and can prevent failures due to clogs or other unexpected behavior. If you can't fix it in design, there are workarounds, but try to need to avoid them!
Avoid Intersecting Shells
As we are at it, often game and graphic designers are lazy and use intersecting shells. These can become quite messy in the slicing step. If possible, try to avoid intersecting shells, even as modern slicers have learned to fix this by themselves by now. The results of that are not always pretty if you forget to flag the &quot;Union intersecting shells&quot; option in your slicer.
Sizing
We might not always be aware of it, but prints do shrink in the XY axis and to a different degree in the Z axis as they cool, during and after the print is done. This is what causes warping in the first place and lead to many lost prints (especially on non-heated beds). This behavior has to be taken into consideration especially when designing bores. My suggestion for this is twofold:

Intentionally design the hole to be too small and add extra wall material in slicing, then drill it up to the right diameter. Drill slowly to not melt the plastic.
Learn your shrinking parameter for the material and design with that shrinkage in mind, possibly iterating the print a few times. Note that different spools/colors of the same material might have different shrinking!

Minimum Wall Thickness
A 3D-printer can not reliably print walls that are thinner than the extrusion width of a printer. The choice of the correct nozzle for an extrusion width is a question upon itself..
Tapping/Screwing/Inserts
For tapping prints directly, you need wall thickness - according to the norms - you'll need usually about 0.2mm diameter that can be tapped into for the standard sizes. Using 3 perimeters with a 0.45 mm extrusion width will give walls of 1.2 mm, which I consider a rather strong wall, and provides quite some tolerance to drill up to size and then tapping screws into. You will get away with 2 perimeters for smaller thread sizes (M3 and lower), but for large ones (M10+), you will want a fourth or even fifth perimeter.
Remember though, that the printed PLA is not good for very strong threading: Tapping prints directly is pretty much only for low- and non-load-bearing connections. If you combine several pieces with screws, try to design the parts to make some sort of compression fit using a bolt and nut, or use several, small diameter screws with a fine thread. Avoid coarse thread if you can, stay on the small side.
If you need a load bearing connection with screws, the best strength comes from using a metal insert or provide a space for a nut to fit into. Metal inserts are usually placed by heat-setting them: put the heat-set insert onto a soldering iron tip and push it into the slightly under-sized hole, melting and molding the print to fit the insert, providing strong threads that are held really good in the shell of remolded plastic.
As a compromise, modern slicers allow to use modifier meshes, that could be used to increase the strength of modeled threads or holes that need to be tapped.
Do you want to know more? CNC Kitchen (Stefan) had made some tests on the strength of these connection type.
Print strength
Keep these general rules in mind when designing load-bearing parts:
Generally speaking, FDM prints are strongest in carrying along their Z-axis when withstanding compressive forces, as then the print layers of the shell are forced against each other. It also excels at fighting bending forces this way. But this orientation is also giving us the lowest tensile strength, as each layer boundary is a possible breaking point.
The XY-plane usually excels in tensile strength but sacrifices some of its ability to withstand compression (it is not proportional though).
Printing a part at a 45° angle will give often a great compromise of strengths, but might need an additional surface to get a good first layer - this surface can be sacrificial with the use of support.
For deeper information on the strength of parts and materials in comparison and how to manipulate it, there are large playlists of tests made by CNC Kitchen (Stefan) and Thomas Sanladerer (Tom)
Post processing
Post processing can be your best friend when printing, just as it can be your worst nightmare. I won't detail all methods of postprocessing, but some that are quite applicable.
Assembly/Gluing
Remember to design your parts with gaps for the glue when designing parts for assembly, and you might want to include guidance notches/noses to make sure the assembly aligns. This is especially needed as the parts shrink a little and have a rough surface.
If you need to assemble your part due to the available print volume, be sure to include ways to key the parts together. Pegs or outcroppings/indents (often called keys) that match up to one another make alignment on assembly much easier. It can be a good idea to design yourself a &quot;cookie cutter&quot; file that is applied after designing the part that automatically includes the glue gaps and keys.
There are a lot of glues and other methods to merge the parts. A more in-depth look at some of them is What glues for PLA? but you will have to keep in mind how you want to combine your pieces in the design step - and account for it.
Print in Place/PIP
In this vein, learn the tolerances your printer can manage to allow print-in-place(PIP), allowing functional parts that require no assembly. PIP is something that isn't possible with subtractive manufacturing usually, but remember that in 3D printing you might need to break the parts free after printing from bridges or sagging. Usually, a single strong turn suffices. To be able to do this, you might want to include a position for an Allen-key to manually turn the parts.
To learn how fine your tolerances are, there are many tolerance gauges/tests around. A rule of thumb for many printers is, that 1 nozzle width is often easily achievable with a good setup, 0.5 nozzles are achievable with some effort and 0.25 is somewhere close to the 'holy grail' - you might want to change the nozzle to a smaller one in case you want to have very thin gaps.
Composite construction
There are ways to turn your (mostly) hollow prints into much stiffer versions of themselves by turning them into composites, for example by using a resin or a different hardening fluid (like foam or plaster) as a filler or coating material.
When planning to do so, remember to include inlets/outlets for it and the air. It can be a good idea to design the part in such a way that it just contains the walls and a pre-planned support structure. In doing so, remember to disable infill in the slicer to enforce the flow you want in your structure. Look at how the ribs inside of an airframe are designed for general rules on hollow parts: include holes. This allows the flow of your fluid into each and every corner instead of blocking the flow. This can also reduce the needed number of inlets and outlets from one per chamber to one per part.
Plastic Properties
Remember we work with thermoplastics. Learn what kind of postprocessing your thermoplastic allows and its mechanical properties. Some examples:

APS can be vapor smoothed with acetone.
Many plastics can be annealed by baking at or little above their glass transition temperature, increasing strength and layer-to-layer bonding.
When using power tools on plastic, use ample cooling and time, as otherwise, one quickly melts the prints!

Surface Finish
The surface of FDM prints is somewhat rough. To smooth it out there are 2 general ways: fill it up or smooth it down. If you want to fill it up, design the part undersized, if you smooth it down, add sacrificial thickness. It is common to combine both, adding body filler first, then sanding down till the print material shows through again. If doing this, make sure to check your sizing.
If there comes a lacquer layer atop, remember to account for that thickness: undersize surfaces, oversize holes!

1 - this means, that thoughts about postprocessing that are not unique to 3D printing are not elaborated on here. Examples are painting, coating or smoothing the surface mechanically.

Further reading/viewing
Further information can be gotten from these playlists, though they aim at times for newbies:

CNC Kitchen (Stefan): 3D Printing for Engineers Playlist
CNC Kitchen (Stefan): 3D Printing Tips
Maker's Muse (Angus): 3D Printint 101
Maker's Muse (Angus): Advanced 3D Printing Tips
Maker's Muse (Angus): CAD for Newbies
Maker's Muse (Angus): CAD for 3D Printing