Does the gravity of the planets affect the orbit of other planets in our solar system?
It does - although the term 'disrupt' may be a bit too strong to describe the effect; personally, I think 'influence' would fit better.
An interesting consequence of such iterations is something called orbital resonance; after long periods of time - and remember that the current estimate for our planet's existence is 4.54 billion years - the ebb and flow of tiny gravitational pulls cause nearby celestial bodies to develop an interlocked behavior. It's a double-edged sword, though; it may de-estabilize a system, or lock it into stability.
Quoting the Wikipedia entry,
Orbital resonances greatly enhance the mutual gravitational influence
of the bodies, i.e., their ability to alter or constrain each other's
Another gravity-related effect (although, as pointed out by Dieudonné, present only on our solar system between bodies that have very close orbits like the Earth-Moon and Sun-Mercury systems) is known as Tidal locking, or captured rotation.
More about orbital resonance on this ASP Conference Series paper: Renu Malhotra, Orbital Resonances and Chaos in the Solar System.
Tidal locking does not actually happen between planets (in the solar system). It only happens when two bodies are very close in orbit around each other such as in close binary stars or between bodies with large mass differences such as planet moon systems where the smaller body (the moon) is tidally locked to the planet. It can also happen between a star and its planets such as Mercury being locked to the Sun.
@Dieudonné, you're completely right; I actually wanted to mention tidal locking as another noticeable gravitational effect, not just between planets. I'll clarify that on an edit. Thanks for the hint!