Off-beat in the British Army

Solution 1:

One of the meanings of beat is ‘the round or course habitually traversed by a watchman, sentinel, or constable on duty’ (OED). If this was extended to military use, then it might follow that a soldier who was off beat was retired, and this use could have found its way into Tamil.

That, of course, is speculation. You perhaps need to look at the question from the other end. Can you rule out any purely Tamil etymology for the term?

Solution 2:

I couldn't find anything to support the claim, although that doesn't necessarily mean it's false.

The 1994 Shorter Slang Dictionary (Partridge, Beale, Fergusson) has another meaning:

off-beat unconventional. Adopted from the USA around 1960.

And the 2008 New Partridge Dictionary of Slang and Unconventional English says:

offbeat adjective unconventional, but not unique US, 1938

This has an origin in music.

Beat also has a policing meaning:

In police terminology, a beat is the territory and time that a police officer patrols. Beat policing is based on traditional policing (late 19th century) and utilises the close relationship with the community members within the assigned Beat to strengthen police effectiveness and encourage cooperative efforts to make a safer community. Beat police typically patrol on foot or bicycle which provides more interaction between police and community members (good and bad).

This is just guesswork, but perhaps there's some confusion with a police officer coming "off the beat" to work in the office or to retire.

Another Chennai Slang webpage says this:

Word/phrase: OB adikaradhu

Meaning in context: To waste time.

Origin, usage: OB is pronounced as the individual letters O and B. Etymology not known for certain. Possibly from "Off Beat", old British military term meaning "off duty". Also possibly from "Out of Business" or from "O'l Bhajanai" (local slang for "doing nothing").