When should you replace tires after getting a flat?
From personal experience, I'd say a larger-than-usual hole in the tire could have these undesireable side-effects:


The tube might get a bit exposed, and the day-to-day rolling over the hole might wear it down until the tire eventually flats out. Chemical aggresion from road grime or mud could also be involved.
The fabric of the tire might get damaged to the point there is a visible bulge. This not only causes a bump while rolling, but most importantly is a weak point that might lead to tire explosion and and accident.


Most of these somewhat larger punctures damage mainly the rubber and not so much the tire fabric (plies). If that is the case, a simple cautionary patch on the inside (with a regular tube patch) is enough to warrant peace of mind.

If the tire is bulging around the hole when it is inflated at the pressure you use, that is a warning indicating the tire is probably too damaged. A larger or stronger patch might be used inside the tire, and some regular observation to see if it stabilizes or grows is needed, but most probably it would be better to get it off the bike (which is very sad when it is a relatively new tire).

Hope this helps