In all three sentences, "everything he needs" is a noun clause which means it follows conjugation rules independent of the main verb, depending on what the intended meaning is.
- Harry stowed everything he needs into his trunk.
This means that prior to now, Harry stowed [things] which he needs in this moment.
It can also mean, that he stowed things that he needs generally (that might be needed at any time). Context determines which meaning is accurate.
Note that, for this latter meaning, you could also use the future or subjunctive tenses "Harry stowed everything he will/could need in his trunk." However, using the future or subjunctive tense only allows for future needs while using the present tense allows Harry to have those needs both in the past and/or in the future because of the generality.
- Harry stows everything he needs into his trunk.
This is similar to the previous sentence's meaning except that Harry is doing the stowing now, instead of in the past.
It also could be a general statement that, from time to time, Harry stowes things in his truck for those occasions he had need of them.
- Harry stowed everything he needed into his trunk.
This sentence means that Harry stowed things in the past for his needs, which were also in the past, but which have been met. Harry no longer has those needs.