Is every sentence in a tense?

To answer the official question, in every sentence (not every clause, but every sentence) the first verb in the main verb phrase must be one of

• a Present tense form (am, is, are, have, has, does, do, or Verb + -Z₃, the 3SgPr inflection)
• a Past tense form (was, were, had, did, or Verb + -ED, the Past inflection)
• a Modal auxiliary verb (can, could, may, might, will, would, shall, should, must)

English modal auxiliaries are not inflected for tense, so they are either not in any tense or they are always present tense, depending on what kind of theory of tense you're applying. So either all English sentences are in a tense, or there are some that aren't. But that's just how one describes the language -- it doesn't affect the grammar.

Time in English is frequently indicated by tense, but often enough it's also -- or even only -- a matter of the words or constructions used. See the Deixis Lectures for more on expressions of Time.

English tensed verb forms, however, often specifically don't refer to time. For instance, the Present tense, when used with an active verb, is most likely to refer to an habitual occurrence than to the present time. E.g,

• Bill walks to school means he walks (almost) every time he goes to school.
• That dog bites means that the dog has been known to bite people on some occasion(s).
• Mary drives a Toyota means that Mary usually drives (and probably owns) a Toyota.

None of these refer to what Bill, the dog, or Mary are doing at the present time -- neither the time of speaking nor the "present" of a narrative. This is called a generic construction.

The particular use in the original question licenses the use of a past verb form to indicate an unreal supposition, much the way certain regular subjunctive verb endings do in European languages; but only sporadically -- not regularly. This counterfactual conditional construction, like most archaic remnants, is idiomatic, and governed by only a few constructions and verbs. So one finds

• I wish I were home now.
• If I were you, ...
• If I had the money now, I'd give it to you.