The most important thing: with a printer this old, compatible cartridges won't break anything that Canon itself hasn't already broken over time. That whole printhead myth is mostly a legend spread by manufacturers to get you to buy cartridges at full price. Decent compatibles (not the cheapest ones, though) work great on old Pixmas, especially on models like yours that are reliable and tough.
What you should do is try well-known brands and not the cheapest possible options: there are quite a few that have good reviews and cost half as much as the originals. If the printer still works well, it'll keep working with compatibles too, at least until the hardware fails from age. In your case there's no reason to spend a fortune on original cartridges.
The only real risk is figuring out which compatible cartridges work well for your specific model - you can ask in forums or check reviews for the MP250 specifically. But no, they won't destroy your printhead just because they don't have the Canon brand on them. If the printer has held up until now, it'll hold up with compatibles too.