Alpha, Beta, and Arabian Nights in Tournaments?
There is a special limitation on Alpha cards to account for their differently-cut edges. From the official Magic The Gathering Floor Rules:

102. Authorized Cards
Alpha cards (cards from the first print run of the basic set) may be used in decks  containing non-Alpha cards only
if all cards are placed in completely opaque sleeves  and only if the
sleeves could not be considered marked. If sleeves are not used, Alpha
cards may be used only in decks that consist exclusively of Alpha
cards.

There are no such restrictions on Beta or Arabian Nights cards however, which are (hopefully) less easily distinguishable by their shape from modern cards.
As rahzark's answer rightly suggests, individual cards may be restricted or banned in different tournament formats, but there's no prohibition against cards from older sets as a whole.