I like Smart_TJ's answer (and I voted in favor), but I think it's just as valid to rule in reverse.
On this topic, Timeless body only declare that
(…) you no longer need food or water
This statement is not limited to a specific category of needs (it does not say "you no longer need food or water for the sole purpose of subsistence"). It is plausible to say that the correct interpretation of this statement is that" you no longer need food or water for any purpose".
From this point of view, the rule of exhaustion would be judged as follows:
- You have at least a level of exhaustion
- You are taking a long rest to reduce your level of exhaustion by
- To this end, you should normally ingest food and
- Timeless body explicitly eliminates the need for food and water for any purpose
- Therefore, the obligation to ingest food and drink becomes
- You can, therefore, remove a level of exhaustion simply by taking a
long rest, as no other constraint or requirement applies beyond the rest itself
It works on two of the most common ambiguity heuristics: things do what they say they do (no need for food), and specific beats in general (when most would need food and drink, a character with Timeless body specifically not).