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Are there languages without any copula?

Of note, Hebrew and Arabic don't have the verb "to be" in the present, but do have it in the past for predicative sentences ("I was happy").

Also of note, if semantic adjectives are syntactically stative verbs, as in Chinese, but a copula is still required for predicative sentences involving nouns ("cats are mammals"), then it's not what I'm looking for.

If a language has no copula, how does it handle verbal inflections (if any), such as tense and aspect? In Hebrew, again, the copula is dropped in the present, but restored in the past and future to convey tense.

Ancient Egyptian may come close to what you're after. There are a few words that approach the copula in meaning and use, but of these, wnn is properly a verb meaning "exist", or "there is/are", and ỉw is functionally a particle used in compound verb forms, and to introduce sentences with adverbial predicates in two cases - firstly when the subject is a pronoun, ỉw is simply used as a word for the subject pronoun, being a clitic, to attach to, and secondly when the subject is a noun, ỉw is used to indicate a kind of impersonal detachment.

With a bunch of caveats, wényán (Old Chinese).

The basic way to say "X is Y" is X Y yě. You might think yě is the copula, and some professors think it is. But it doesn't appear in a verbal position, it can't be modified or negated, and it can be used with other verbs. It's basically an evidential (asserting a fact) with a side helping of aspectual (it contrasts with yǐ which marks a change). There is no tense, and aspect is indicated with particles anyway.

Caveat 1: there is a negative copula, fēi.

Caveat 2: The verb wéi "make, work, serve (as)" sometimes comes close to a copula. After all there isn't much semantic space between "he serves as a minister" and "he is a minister". Also, there was a copula in very early stages of the language (another wéi).

The Mandarin copula shì derives from OC "this".

I think there are a whole bunch of languages that don't necessarily have anything that can be called a copula and also a whole bunch in which the copula sees limited use. IIRC, Malay/Indonesian technically has a copula adalah, but it isn't used much in the spoken language. It also has no verbal inflections. I think a bunch of Austronesian languages (maybe even all of them?) also lack a copula and do a number of different things instead, e.g. Tagalog uses a "linking verb" ay, but this is not only used where one would use a copula in English, and Malagasy I think puts tense prefixes on elements that are not verbs such as adverbs or prepositions. (Take all of this with a grain of salt. I'm not an expert on any of these languages and don't know any of them too well).

If you have TAM material as distinct words it's easy enough to have zero-copula and still distinguish TAM. I'm pretty sure this crops up a lot in Southeast Asia.

Nuu-chah-nulth doesn't have a copula. The compliment just takes all the inflectional material (except maybe aspect?), as most of it is in a huge clitic chain attached to the first element. The exception is equatives, where a relevant pronoun takes the inflection instead (in sister language Makah there *is* a copula, but only here - all persons use a dummy 3rd person pronoun to carry inflectional material in equatives). A Makah example pieced together from a grammar:

[hʊχ'tʌkʰsa:qʰtɪʔi:ʔɛyɪka:kʰtɬʔɔ]
huχtak-sa:q-tiʔi:='eyik=(q)a:k=tɬʔo
know.how-CAUS.PERF-...er=FUT=POLAR.2S=again
Will you be a teacher again?

As far as I can tell with a glance, Salish (or at least Halkomelen) is similar that nominal predicates can inflect for past tense directly (other TAM material appears to be particles), though it does have a copula in clefting.

It seems pretty common to me, though, that nonverbal predicates may just not allow the full range of TAM expression. Mayan, for example, allows nouns, adjectives, and other elements to act as predicate heads directly, but at least in Ch'ol they are incompatible with aspect marking despite aspect marking being distinct words and not inflections.

If you're interested in more languages that work like Hebrew and Arabic wrt the copula, Russian also has a zero copula in the present but has an inflected copula in the other tenses.

BSL and (probably) other sign languages have zero copula

Okay, so putting TAM markers on non-verbs acting as predicates can be done?

Of course all of this gets a lot easier in analytic languages, because then you don't need to worry about nouns taking verbal morphology and such. Of note, Hebrew, Arabic, and Russian are highly inflected, and don't put verbal morphology on nouns, although apparently Malagasy and Salish do.

In form and inflection, the Irish copula has a great deal in common with preverbal particles and almost nothing at all in common with actual inflected verbs. It's probably best to analyse it as a particle which permits predication of a following NP.

I think there's a difference between "lacking a copula" and "having a copula that is not a verb."

I'm confused about what you thought I said. I only said that I think Malagasy puts tense prefixes on things that are not verbs including adverbs and prepositions. I never said anything about nouns there.

"Mali is my cat" would probably look something like 'Mali cat-my', or 'cat-my Mali', or there may be a focus marker in between 'Mali' and 'my cat'.

In Dothraki, which has no copula, the present tense is Mali havzi anni “Mali cat my”. In the past tense you would say (Jada) Mali havzoon anni, which puts havzi “cat” in the ablative case, such that the word-for-word translation is: “(Comes) Mali cat-from my”. And yes, this is also how you say “Mali is/comes from my cat”. You specify which you mean with jada “comes”, which is placed sentence-initially for some godforsaken reason in copulative sentences, thus: Jada Mali havzoon anni “Comes Mali cat-from my”, as opposed to Mali jada havzoon anni “Mali comes cat-from my”, which means “Mali comes from my cat”. It only gets worse when you realize that the allative case (Ee) Mali havzaan anni, with or without the sentence-initial verb ee “goes”, means “Mali will be my cat”, and Mali ee havzaan anni means “Mali goes to my cat”. I only hope you find something more elegant.