Not loads, but more than we did 20 years ago. We know it's Indo-European, that's not really up for debate at this point. Almost certainly Italo-Celtic. It had at least to some extent Latin influence, as well as what was probably a Celtic substratum. Some people (Witczak being one) argue for it as a Pre-Celtic language (consistent with Italo-Celtic anyway). I don't believe we have anything conclusive on that particular point, and my understanding is that we can't actually say for sure that it's one thing or another at this stage. A lot of similarities can be explained as a result of language contact, as is the case with a lot of now-gone languages.
The cultural practices were in line with Celtic practices, and this has given support to it as a Celtic language, but the influence of Latin was substantial, and so there have been arguments for it as something related to Latin. One difficulty in this is that without a substantial data corpus, it's really hard to drill down and work out relatedness of languages. You need more than vocabulary, although some have made (generally poor and unconvincing) attempts at categorising languages into specific attested families entirely through the use of toponyms.
Ideally you need a lot of examples of full sentences in context from which to draw out the syntax of the language, and from there you are able to make better deductions as to genetic (in the linguistic sense) relatedness. Unfortunately, for Lusintanian, we have a handful of inscriptions, and the ones which have considerable Latin are already a little suspect as being potentially more influenced than the language might have otherwise been pre-Latinisation. Regardless, there are a number of morphological correspondences, so this all seems quite reasonable a Italo-Celtic anyway.
The known vocabulary is small, the known grammar is limited. We can't reasonably reconstruct too much, as we don't really have any descendants from which to do that. Chances are we will continue to not know terribly much about it unless further inscriptions are discovered (a definitely possibility) which give us more information than what we've got already.
Full disclosure, I myself am not an Indo-Europeanist, but am a Historical Linguist. I welcome additions/corrections where needed.
See also:
Language contact in the pre-Roman and Roman Iberian peninsula: Direct and indrect evidence by Simkin
Religion, language and identity in Hispania: Celtiberian and Lusitanian rock inscriptions by Alfayé and Marco-Simón
On the Indo-European origin of two Lusitanian theonyms by Witczak
Some notes on Lusitanian by Wodtko.
Much like /u/keyilan has written, Lusitanian is definitely an Indo-European language, and probably influenced by Celtic, but its exact position within the family is unclear. It is likely that it is either an independent dialect unrelated to any existing IE language, a distant relative of Italic or may have been part of another sub-branch closely related to Italic and Celtic whose other members are unattested.
Essentially, despite the fact that we don't have a good grasp of Lusitanian vocabulary, we can still identify that it is an IE language due to its grammar, whose IE inflectional cases are evident in surviving inscriptions:
RVFINVS. ET TIRO SCRIP/SERVNT VEAMINICORI DOENTI ANGOM LAMATICOM CROVCEAIMAGA REAICOI. PETRANIOI. T ADOM. PORGOM IOVEAI CAELOBRIGOI (this inscription dates from the 1st century AD, explaining why the first four words; "Rufinus and Tiro wrote this", are in Latin)
Although the definitions of many of the words in this inscription are uncertain, we can still tell that it is some sort of offering made to a god or gods (most attestations of early Celtic and related languages take this form), addressing the deities and briefly describing the offering made to them. What we can discern is that two men, Rufinus and Tiro, who belonged to a people named the Veaminicori (we can tell this because it's a noun inflected in the nominative plural) who have given ("doenti" cf. Latin donus/bonus, Proto Celtic *danu-) a grouping of animal sacrifices, identified by their accusative singular inflection, to the "Ioveai" and "Caelobrigoi" (whatever that means), which are inflected in the dative singular. These are most likely the names of some Lusitanian deities.
This inscription alone helps disprove Lusitanian's placement in the Celtic language family due to the appearance of the word "Porgom" - I mentioned that this inscription describes some sort of animal sacrifice, as "porgom" and "angom" (the only really identifiable words in that phrase) likely mean "pig" and "lamb" respectively (cf. Latin porcum & agnum).
The Celtiberian dialects of the Celtic languages split very early off and preserve a phonological feature of Proto-Celtic that is preserved only in the Goidelic family; the loss of Proto-Indo-European *P, which was only regained later by Gaulish and the British Celtic languages by assimilating Proto-Celtic *Kw > P. This is one of the most easily identifiable and characteristic sound changes of the early Celtic languages. If Lusitanian were a Celtic language that spread into Iberia just before or at the same time as Celtiberian, we would expect to see the same loss of PIE *P, but this is just not the case. As well, Proto-Celtic seems to have replaced its word for "pig" with a borrowing from some non-IE substrate (*mokko-, or something along those lines), further distancing Lusitanian from the Celtic family given the presence of an IE inherited term "porgom" for "pig" (presumably).
If Lusitanian were initially descended from Proto-Celtic, we would expect the word "porgom" to lack its initial P-, but given its presence here and in other inscriptions, its placement within the Celtic language family is doubtful. What's more likely is that it was descended from one of the many dialects that formed a Sprachbund in Central Europe before Italic and Celtic languages spread southwards and westwards respectively (I also do not buy the theory of a unified proto-Italo-Celtic language) and developed into more distinct language groups. So the most likely explanation, IMO, is that Lusitanian was some dialect that had developed alongside Proto-Celtic and Proto-Italic in Central Europe and split off before becoming influenced by its two overshadowing cousins. At the same time, it seems to bear a closer resemblance to Italic than Celtic, which may suggest that it was a very early dialect of Italic or even Proto-Italic that split and found its way in Iberia before those languages even penetrated the Italian peninsula.