Title says it all - does the shape of a wave have any effect on how we perceive it as a sound?
Yes, but not directly - instead, we perceive the different amounts of energy in different harmonics. :)
A square wave only has odd harmonics (the base frequency, 3x the base frequency, 5x, 7x, etc.) and that gives it a distinct character. This is similar to the triangle wave (which also only has odd harmonics, but in different amounts). They are both very tonally different to the sawtooth wave, which contains both odd and even harmonics.
However, since it's mostly determined the relative amplitude of the harmonics (we aren't very good at hearing phase, except between our two ears) it's possible for two waveforms to look different but sound basically identical, if their harmonics have the same energy.
Here is a video which demonstrates that: at 0:02 it plays a conventional sawtooth waveform, but at 0:12 it plays a different-looking waveform which has the same energy in its harmonics, so sounds the same.
Edit: as @MacTuesday mentioned, the distinction between a continuous tone and a quickly-repeating sound can be ambiguous, particularly for very low pitches. Phase/timing information is not meaningless, but for the cases you're thinking of, harmonic energy is the primary aspect that determines timbre of continuous sounds.