Sometimes I hear people geminate consonants after stressed vowels in speech. For example, amāta might be pronounced as amātta. I have not heard enough to tell if this gemination is consistent for those who do it, or if it requires that the vowel is long, too. My understanding is that this is not valid classical pronunciation. But was this kind of elongation standard or acceptable pronunciation at some later point in time? Such gemination seems to happen in Italian: It sounds to me that amata has a long -t-.
You're right that such gemination is not correct Classical pronunciation, and I believe the answer to your question whether it occurred in post-Classical Latin/Romance is no. Italian amata does not, at least in standard pronunciation, have a geminate; phonetically it's [ama:ta], versus e.g. matta [mat:a] with a geminate t. To the best of my knowledge, there are no instances of innovative gemination after Latin stressed vowels in the Romance languages.
There is, however, a related phenomenon which seems to have taken place in the prehistory of Latin, known either as "the littera rule" or "the Iuppiter rule", both those words being examples of it. This is a highly debated sound change and the details are controversial, but basically, with certain words we see variation between a long vowel followed by a single consonant (lītera, Iūpiter) and a short vowel followed by a geminate (littera, Iuppiter). In most or all such words the vowel is in the initial syllable, which would have always borne the stress in earlier Latin (and in most such words the initial syllable is still stressed in Classical Latin), so it's possible that the gemination is related to stress. For a recent overview see this handout by Michael Weiss.
Another somewhat similar, though definitely post-Latin, phenomenon is so-called raddoppiamento sintattico (syntactic gemination) in Italian. This doubles the initial consonant of a word when the preceding word ends in a stressed vowel: e.g. in parlò bene "he spoke well" the b is geminated, but not in parlo bene "I speak well" where the preceding o is unstressed. But this is an Italian development, not an inheritance from Latin.
I don't know exactly why you have heard pronunciations of Italian amata with a long /tː/, but I would guess this is just a case of different speakers using different phonetic durations for phonemically singleton consonants. The exact realization of gemination is not the same in all languages/language varieties, so some Italian speakers may use something that sounds like /tt/ to you, but that sounds like /t/ to them (and that is distinct from their /tt/). I think that's what's going on in this case. Unfortunately, I haven't been able to find a study that directly compares the phonetic correlates of phonemic consonant length in Finnish and Italian.
Since we don't have access to ancient Latin speakers, we don't know much about the phonetic details of how they pronounced geminate consonants. We only know that there was a phonological distinction between geminates and singleton consonants, and that there was also a distinct phonemic contrast between long and short vowels (Latin vowel length, unlike Italian vowel length, was not completely conditioned by the phonological context). The way modern Latin speakers pronounce geminates is probably heavily dependent on their L1s.
In addition to what TKR said about the "littera" rule (which is about cases of alternation between long vowel + singleton consonant and short vowel + geminate consonant that seem to date back to early Latin), there were some cases where a singleton consonant of Classical Latin developed into a geminate in an ancestor of some Romance variety, but there doesn't ever seem to have been a sound law of the form "VːC" > "VːCː".
One isolated example that I know of where a consonant appears to have been lengthened after a stressed long vowel is French tout, Italian tutto from Latin tōtus. The exact explanation of these forms seems a bit disputed, but the CNRTL says this came from a form "tōttus" that was called a "barbarism" by the grammarian Consentius. According to the CNRTL, it's an example of "expressive gemination", which is not a regular sound change. The Weiss handout that TKR's answer mentioned (Observations on the Littera Rule") seems to treat "tottus" as an example of the "littera" rule, but I'm not sure if it is quite the same because I don't think Latin ŏ/Proto-Romance *ɔ would be expected to evolve to French ou /u/.
Quantity and Quality in the Vowel-System of Vulgar Latin, by N. C. W. Spence (2015) says
It is worth noting that the early examples of vowel reduction in combination with consonant groups (cŭppa alongside cūpa, bŭcca alongside būca, etc.) are not on a par with the reductions exemplified by stēlla > stẹ̆lla, tōtus > tọ̆ttus, Italian villa, mille, anguilla, zitto, etc.: the ŭ of cŭppa and bŭcca has merged in W. Romance with the continuant of Classical Latin ō, whereas the reduced i of villa, etc., has not merged with the continuant of Classical Latin ē. That is to say that the vowels in the words of the second group were adapted to a system in which vocalic quality, rather than length, was already distinctive.