What does “there” mean in the last sentence?

Yes, it appears you are reading it correctly.

"There" appears to refer back to "hers" which refers back to his...sunburned hands." So the warmth is there, in her hands, and it filters (moves as if through a membrane) to his hands.

There is a subtle inconsistency too: "hers" refers to "hands" (plural), but the warmth filters from her "hand" (singular) to his.

I don't agree with the preceding answer and I think that the adverb "there" in this context is to be given the meaning in OALD, 4, that is,

• "at that point (in a story, an argument, etc.)".

If we were to replace this adverb by an equivalent we would then write

• "And as if listening carefully to something, he felt at that moment the warmth filter from her hand to his."

or

• "And as if listening carefully to something, he felt the warmth then filter from her hand to his.".

"Filter" is used without an s because it is not the main verb in the verbal form and it is in fact an infinitive form (not conjugated, s is meaningless then). the verbal form is "to feel sth do sth (OALD, 2).

• feel somebody/something/yourself do something
  I felt something crawl up my arm. We felt the ground give way under our feet.

he felt the warmth there filter from her hand to his

In the sentence, there pinpoints where his hands engaged in physical contact with her hand.

why “filter” verb is used without suffix -s?

There is a verb pattern hear, see, etc. + object + infinitive or -ing.

One can use either the infinitive without to or the -ing form after the object of verbs such as see, hear.

The infinitive without to refers to the whole event someone hears or sees.

For example, "I saw him go." I observed the whole event.

The -ing form refers to an event that is in progress.

For example, "She heard them talking." The event was happening, not completed.

In your example, the person observed the whole filter event.