"Tapping" an address/bus line
You could use a latch chip to "sample and hold" the data lines.

Something like the 74HC573 might do the trick.  Connect your data lines to the inputs of the latch chip and connect your LEDs to the outputs.  Then have a way to toggle the latch pin on the 74HC573 at a slow rate (perhaps 1 Hz would suit your purposes) to refresh the display.  You can simply tie the ~OE pin ("Output Enable, active low") to 0V.  You can probably find a slowly toggling signal in the system somewhere or you can generate one yourself with something like a 555 timer.  The latch chip's datasheet will show a diagram of the pulse (duration and direction) that you need to send to the LE pin ("latch inputs, active high").

If you didn't have a way to sample and hold the values, then it is likely that the diode would simply appear dimply lit all the time, which would look unimpressive.  Toggling the refresh of the display slowly allows each LED to light up brightly and gives the human eye time to see it clearly.

The 573's inputs have a high impedance and therefore will not adversely affect the bus lines you are interfacing with.  Although you should be careful to keep your "tap" wires a short as possible to avoid interference.

Your concerns are valid, placing LEDs directly onto these lines would likely cause all kinds of problems and may potentially damage the device.  This is because data / address lines are not intended to be power sources to high current devices like LEDs, they are intended to be voltage-only signalling.

Latches are one of the simplest chips to use and will make a fun project for you I'm sure. :)  573's often have a nice wide voltage range (2 to 6 V) which makes life easier when playing with older computers.