Can you damage a tantalum capacitor by mounting with heat gun?
Any component can be damaged by too much heat.  Factories carefully control the temperature of their "solder reflow ovens" to ride that balance between properly soldering the part and destroying it.  It is more than just temperature that they control, it is the time or speed at which a part is brought up to a temp, how long it stays there, and then how quickly it cools.  Some ovens have 7 or more zones, each with their own temperature controls, just for this purpose.

The problem with a heat gun is that you have no real control over anything.  While the gun might be set for 325 deg C, you have no idea how hot the parts were getting, or how fast.  Could the heat gun have damaged the parts?  Absolutely.  But it could also have worked, too.  

There are these things called "hot air reflow solder stations", which is essentially an expensive heat gun designed for doing things like this.  It has lots of controls for soldering parts without damaging them.

When it comes to caps, there really isn't any reason to use a heat gun-- or even a hot air reflow solder station.  There are only two pins, which are relatively big.  Even if the PCB was done wrong, there are always ways to hack a cap on there.

The only parts that I can't solder with a standard soldering iron are BGA's, and some QFN's with the E-Pad on the bottom.  Everything else, TSSOP's, TQFP's, etc., I can just use a standard iron.