It depends on what you mean by Latin and what you mean by dead.

For example, Classical Latin isn't actually the ancestor of the modern Romance languages. Technically, that's Vulgar Latin. In short, just like with English, there was the standard language used in formal situations, but there were also all sorts of common mistakes made in common speech.

As an example of this with a parallel in English, we can look at verbs. Germanic languages tend to have two ways to form the past tense. Some verbs, called strong verbs, change vowels in the past, like write > wrote, while others, called weak verbs, add a -t or a -d, like love > loved. Weak verbs are easier and include most borrowed words (strive/strove is an interesting and notable exception), so there's a trend over time for verbs to become weak. For example, the King James Bible features "crew" as the past tense of "to crow", but today we'd just say that cock crowed three times. Similarly, the Latin perfect conjugation was... complicated. Sure, if you look at first conjugation verbs, most or many of them use -avi, but there were also words like "lavare" just being "lavi" or "secare" being "secui". Vulgar Latin, though, didn't care. People just generalized -avi to all those cases because it was easier and the most common. We see this in modern Romance languages, like how Portuguese "segou" eventually comes from a proto-Romance *secavit, not "secuit" like would have been grammatically correct.

There were also pronunciation shifts. For example, Latin originally distinguished /b/ and /w/ as different sounds. But /w/ was already starting to be pronounced /β/ (sort of like English V) by a lot of people, and /b/ was shifting to the same sound, though only between vowels. This actually led to other problems, like the future and past tense being pronounced nearly identically, which probably contributed to the adoption of the "infinitive + habeo" future, which itself led to the future tense endings in many Romance languages resembling avoir/haber/avere or whatever the local reflex of "habeo" is.

By the 3rd century, you get the Appendix Probi, which was full of corrections like "auris non oricla", the Latin equivalent of "You can't feel nauseous. You feel nauseated". So was Latin still spoken then? Sure, but it wouldn't have been Cicero's Latin.

This also gets into the second issue. What does it really mean for a language to die? Sure, Latin's still being used by the Catholic Church in a vaguely classical form (there are a few syntactic differences, but I don't think they're relevant). But in a way, it's also still being spoken in Spain and France and Portugal and Mexico and Brazil... At what point does a language cease to be itself and become a new language? Clearly it happens at some point. Like I don't think there are any English speakers who could instinctively understand Old English, like "Hwæt! We Gar-Dena in gear-dagum / Þeodcyninga þrym gefrunon / Hu þa æþelingas ellen fremedon". Meanwhile, Shakespeare is mutually intelligible enough that for as much as people like to complain, it is considered the same language. For example, possibly except for "suffer", every word used in the lines "To be or not to be; that is the question / Whether 'tis nobler in the mind to suffer / The slings and arrows of outrageous fortune / Or to take arms against a sea of troubles / And by opposing end them" is used with the modern meanings.

Or what about languages branching? Scots is sometimes considered a distinct language, also descended from Middle English. But it's also mutually intelligible enough with Modern English that not everyone does. Or what about American and British English? They're the same language now, but what about 100 years from now? Sure, we can understand each other now, but what if L-vocalization spread throughout Britain, then underwent sound changes? Suddenly, American might say "He's a bad apple", while British would say "Ee's bayd ehpoe". At what point did French and Spanish really become distinct, then? For instance, you might talk about Proto-Western-Romance as Vulgar Latin as spoken in the western parts of the Roman Empire, but at what point does that become a distinct language from, say, Proto-Romanian, and not another dialect of the same language.

So at what point did Latin cease to be a unified language? Well we have rough dates like the 8th century for Old French or the 10th century for Old Spanish. But compared to something like Gothic, which doesn't have any descendants still spoken today, I find it hard to say whether Latin can truly be called "dead".

While we are waiting for a response, you might enjoy reading this post from /u/YuunofYork that proposes a linguistic definition for what it means for a language to be dead. You might syncretate it with this post from /u/SheepExplosion that describes the 9C reforms that put Latin into that state by redefining it.