Does T become S sometimes in casual American speech?

To get simple things out of the way quickly, no, in current American speech there is no general trend towards changing word initial /t/ to /s/.

The sound that you hear in that song is probably literally a slip of the tongue or a peculiarity of that speaker for just that word. The singer does make a similar sound with the same word once later in the song, but doesn't do it with any other instances ('typical' and 'Tuesday' don't have the fricative onset that sounds like an 's'). That is, she doesn't seem to always change /t/ to ... that weird thing she does that's sort of like /st/ but maybe /t/ and /s/ at the same time.

Also, more to the point of the question in the title, it just isn't a pattern in Modern Standard American English or really any other variety of English (BrE, IrE, ScE, AusE, IndE, NigE, SiE, etc etc or any subdialects of any of those).... that I'm aware of.

However, this is not to deny the plausibility of such a direct sound change of /t/ to /s/. In ModE, word initial unvoiced plosives (p,t,k) are all aspirated, and /t/ is articulated in the same place as /s/ (dental or alveolar), and nearby articulations/perceptions often get changed into the other. Also, historically, the High German Consonant Shift led West Germanic (parent language of English, Dutch, and German) to change intervocalic t to s for example eat/eten/essen, water/water/wasser.

So there is a possibility that word initial /t/ in general -could- change to /s/, but there is no evidence that it is doing so in any variety of Modern English.