All other things being equal (build quality, low dispersion elements, etc.) are there any significant differences between fast (f/1.8-2.8) and slow (f/5.6 and slower) lenses when shooting with a f/11 or smaller aperture?
Overall, any modern lens performs well when used in F11. If the model is recent and the price is much higher, F2.8 lenses should perform better in flare and geometric distortion compared to version F5.6. But there are cases of old F5.6 lenses with excellent performance, such as the venerable 400mm F5.6L.
You say f11 or smaller. As a general rule (I have not studied every lens), lenses with very large apertures do not have as small of an aperture. I believe most f1.4 lenses have a smallest aperture of f16. Most 2.8 lenses have a smallest aperture of f22. Most f4 or f5.6 lenses have a smallest aperture of f32. I may be slightly off or may be generalizing too much, but I think that's how it works?
Generally speaking lenses with a large maximum aperture (F2.8 and wider) will have more focus points available and you will benefit fully from any cross type or dual cross type focus points that your camera has.
The manual for my camera (5D mark 4) devotes a whole section to this and classifies every Canon lens that can be used on this camera into one of 11 groups according to the number of focus points that will be available.
Only lenses in group A are able to use all 61 focus points, and all the cross type focus points and dual cross type focus points work to their full potential.
These lenses all have a maximum aperture of F2.8 or wider.
So the simple answer to your question is yes - focussing will be affected if you use a lens with a relatively small maximum aperture even if you are using a small aperture such as F11 or less.
At f/11, there is not a great deal of difference between lenses.
Even at f/8 there is little difference.
Here is a example of test results from a low cost 17-85mm EF-s kit lens that was not highly regarded versus a 100mmL, a much more expensive fast prime. At f/11, the 17-85 actually has a little more resolution in the center, but less at the edges.
If you are shooting at f/11, use a simple and less expensive prime lens.
Thanks for the answers everybody. For work I take a lot of sports photos in dismally dark high school gym while I take wildlife photos in full sun for fun. That's what made me think about this question. Well that and the price difference between f/2.8 and f/4 zooms. (I use the Tammy 150-600 for birding, f/5-6.3 too)