Effects Compatibility - What works with what?

There aren't many built in effects incompatibilities, but there are some. The places that could cause you some trouble are:

Low input impedance: When something has a low impedance, that's another way of saying that it takes a lot of current to raise much voltage there. There's one good example here - the Fuzz Face and its clones and imitators.  Most effects have an input impedance that's high enough (say, over 10K ohms) that most any effect can drive them

High output impedance: This is the reverse of a low input impedance - when an effect has a high output impedance, it is easy to load down because it can't put out much current. There aren't many effects around that have this problem, fortunately. This is a problem in vacuum tube circuits, however. Loading of tube outputs can cause serious loss of signal.

DC levels at input or output: In a perfect world, every effect would naturally have a 0 volt DC level at its input and output, and we wouldn't have to use capacitors to block the internal DC levels, nor pulldown resistors to hold the capacitors' outboard ends to ground. Alas, Mother Nature does not run the place that way. Some very few effects "designers" ignore the problems that DC levels at inputs or outputs can cause, and leave out the DC blocking capacitors. This makes for very odd problems when something like a low impedance guitar pickup is hooked to the input and this changes the input bias, or when an effect has a DC level on its output and is connected to the direct coupled input of a tube amp.

Signal level matching problems:

Some effects produce a larger or smaller output signal than they got in. It's rare to get one that's smaller, but it does happen sometimes. Lots of them produce much larger output signals than they get in, distortion pedals and "boost" pedals being the worst offenders. This may cause problems by overloading the input of the next effect in line. 

While overdriving tube inputs often sounds good, overdriving a solid state effect input usually sounds awful. This is especially the case where the effect being overdriven is something like a delay. This unintended distortion is not the carefully crafted clipping of a distortion pedal, but the harsh limiting of silicon at its worst. The solution is pretty simple - turn it down; all effects that can make big signal levels have volume controls.

Another signal level problem can come with putting effects in amplifier effects loops. The signal levels here are usually at least ten times the signal level from a guitar, sometimes much more. Effects loops don't always have level controls, although they should. Just be aware that not all effects will work in effects loops from amps.

Phase Inversion: An effect should always put out signals in phase with the signals at its input. While the human ear is not sensitive at all to absolute phase, any time you get two amplifier channels going out of phase, they may cancel. An inverting effect will have no problems until you go to setups with more than one audio path - say to two amplifiers - and the inverting effect is in only one path. The amp outputs will then partially cancel acoustically. This is always worse in the bass notes, so you may just hear a loss of bass.