I have a Ground Fault Interrupter in the Kitchen and Bathrooms. They were installed in 2006.
When they are tested they do not trip off. I have an outside light, which is switched off because a tree branch keeps turning the light on. Could this switch being turned off cause the
GFI's to not trip properly?

No. How are you testing the GFCI for the OFF condition?

If you are pushing in the "test" button, you may not be pushing it in far enough. If you are pushing it in far enough to trip and it doesn't, the GFI is bad. Best to use a circuit tester with a GFI test button.

Had a home inspector who used a tester and it apparently did not trip using it.
Just thought it could be a switch inline with the GFI may have prevented it from doing what it should have.

The counter GFCI receps. In the kitchen should not extend beyond the kitchen - dining room.

The only way a switch could interfere with the operation of the GFCI receps. Is if it is on the supply side of the GFCI circuit.

Did a little reading on GFI's needing to be tested monthly. These have been in for at least 10 years and more than likely not even having anything plugged into them for more than a week per year. Will check them when I get down there this week.