Yeah, the focus from those against him was more on his Romulan heritage and not that he had lied on his applications. Which I suspect wouldn't get him too terribly in trouble especially if Picard put in good words in Tarsis's behalf. I mean he'd get something on his record for sure and may see a suspension or something of the like, but it's unlikely his career would be over. If there's anything consistent we've with Starfleet when it comes to their reaction to people who break the rules it's that SF has a very "if the ends justify the means" type of attitude.
Starfleet may very well see the "point" behind Tarsis's lie and understand why he would make such a lie and be forgiving towards him. We could also probably quibble that it's only a lie "on paper" since Romulans and Vulcans are, technically, the same species and are just different cultures even if there may be some minute genetic differences between the two given the length of the separation.
It's like telling someone you're Chinese instead of North Korean. Different cultures, maybe some tiny genetic differences, but still the same species.