If I leave fully-cooked food (particularly meat) out at warm temperature - say on the counter or in a crock pot that's been turned off - how long will it stay safe to eat?
Does it make any difference if I re-cook the food afterward?
The USDA has this to say on it:
You'll find similar statements from government agencies around the world. The safe limit for raw or cooked food is 2 hours in the danger zone (40-140° F or 4.4-60° C).
If you're a restaurant owner or cook, you must follow this rule, hold hot foods above 60° C and quick-cool other foods before refrigerating. If you are not working in a professional capacity then you are not legally required to follow it, but if you are serving guests then it would be irresponsible (and possibly actionable, if someone gets sick) to do otherwise.
If you're an individual serving only yourself, then take whatever liberties and break whatever rules you want; it's your food, and your body. But there's no table or chart anyone can give you; there's no single specific point at which a food transitions from "not entirely safe" to "probably will kill you" because it depends entirely on the food, the environment, your immune system, and a plethora of other variables. The rule is 2 hours, period; any longer and there is some non-trivial risk to your health.
Some hints, tips, and warnings:
UK Food Hygiene Regulations (see UK FSA web-site) state that cold foods must be kept at 8°C or below and hot foods must be kept at 63°C or above. This is a legal requirement throughout the UK.
However when you are serving or displaying food, you can keep it out of temperature control for a limited period of time: Cold foods can be kept above 8°C for up to four hours. You should only do this once. If any food is left after this time, you should throw it away or keep it chilled at 8°C or below until it is used. Hot foods can be kept below 63°C for up to two hours. You should only do this once. If any food is left after this time, you should throw it away, reheat it to 63°C or above (82°C in Scotland), or cool below 8°C This applies to the UK with relatively temperate ambient temperatures - there have been nasty food poisoning cases where poorly prepared foods have been held in hot cars for relatively short periods.
Having said this, cold foods should always be served cold as soon as possible, and hot foods served hot as soon as possible after preparation.