Here is an Ngram chart showing the trend in published usage of 'tis over the period 1800–2005 in American publications:
Even the very substantial long-term downward trend that is evident in this chart greatly overstates the extent to which 'tis remains alive in American English today. Most of the matches from recent years of publication are reprints of classics from England ('Tis Pity She's a Whore), reprints of classics from the earlier years of the United States ('Tis Sixty Years Since), allusions to old songs or other quotations from England ('Tis the Season) or the United Sataes (America My Country 'Tis of Thee: An American Song about Freedom), relatively recent books by Irish or Irish American authors ('Tis Herself: An Autobiography), and recent novels attempting to sound authentically archaic (Mystery in the Medieval Castle).
So when you remove the old stuff, the anachronistic new stuff, and the non-native American interlopers, you're left with a current rate of un-self-conscious modern use very near zero.
'Twas not ever thus. If you go back to the age of Emerson in the United States, use of 'tis might already have been mannered, but it was still quite common. Thus, in "Fortune of the Republic" (1878), Ralph Waldo Emerson writes,
The end of all political struggle is to establish morality as the basis of all legislation. 'T is not free institutions, 't is not a democracy that is the end,—no, but only the means. Morality is the object of government.
But at some point (or perhaps gradually) in the nineteenth or early twentieth century, everyday U.S. English seems to have rejected 'tis in favor of it's—still a one-syllable contraction, but with a different i dropped out of the source phrase "it is."