Out on a tear last night - meaning?

"Being on a tear" in 19th, early 20th century (and possibly current) lowland Scots dialect meant being out having a raucous, and probably drunken good time. My source for this is the traditional song The Day We Went to Rothsay, O!.

The song tells the story of a group of men in Glasgow on a public holiday going down the Clyde to the resort of Rothsay on a paddle steamer. Various unruly things happen and in the last verse they are sent back to Glasgow by the police

The Polis wouldna let us stop

Another nicht in Rothsay O!

The first four lines of the song are

One Hogmany at Glesca Fair,

There was me, mysel' and sev'ral mair,

We a' went off to hae a tear

An' spend the nicht in Rothesay, O,

Whether "On a tear" was ever exclusive to Lowland Scotland or whether it was also Irish, English or even, originally, American I don't know; but even if it was exclusively Scots at some time it is very understandable that it would have crossed the Atlantic with migrants.