Euler proved that every even perfect number is of the form $p(p+1)/2$ for $p$ a Mersenne prime, in particular it is equal to the $p$-th triangular number (the sum $1 + \ldots + p$) for some Mersenne prime $p$.
Of odd perfect numbers it is not known whether they exist but lots of constraints (number of different prime factors etc) on numbers of this sort have been proven over the years. My question is as in the title:
Is it known that odd perfect numbers are necessarily triangular or is there no known restriction on odd perfect numbers of this form?
I think a simple answer to your question is: we don't know. If any odd perfect number does exist, we would then see if it's also a triangular number - there are no results that establish this question a priori.
However, if one does exist it raises the question of whether there exist finitely or infinitely many such numbers, and afterwards the question of whether all of them or not all of them are triangular.
If odd perfect numbers do not exist, then no odd perfect number would be equal to a given triangular number.
Going back to Euler's proof, it talks about even perfect numbers.