Eugene, OR
Yes, indirectly. When gutters are blocked, water that's supposed to be carried away through the downspouts ends up pooling around the house instead -- and water sitting against a foundation over time is a well-known path to foundation problems.
The gutters only solve half the problem -- they have to actually drain through clear downspouts, or the water just backs up and finds the nearest low point, often right next to the foundation.
Keeping gutters clear on a regular schedule costs far less than addressing foundation or siding damage after the fact, which is the main reason it's worth staying ahead of rather than catching up on.