No effect, on my server, a $5-per-month Digital Ocean droplet. Using the program called Apache Bench (ab) to retrieve a file four directories deep: /a/b/c/d/e.html

With .htaccess files turned off (AllowOverride None), I got 282 requests per second. With .htacess files on (AllowOverride All) I got 286 requests per second (Yes, actually a few more hits per second with .htaccess turned on, but the results swing from test to test by a few percent anyway).