What is the origin of the phrase "do a line with someone"?

The other answers here refer to the drug taking aspect of it, but I've often heard my Irish father say "doing a line" in reference to dating. It's no longer commonly used in Ireland any more, but most Irish people would know exactly what you meant if you said you were "doing a line" with someone.

This site has a section on Irish Sayings for St Patrick's Day which attempts to explain the etymology of the expression:

I would just like to firstly clarify that ‘Doing a line,’ has nothing at all to do with the use of any illegal substances whatsoever. Instead it was a perfectly innocent activity that young Irish couples partook of as part of a rite of passage to becoming a respectable married couple.

‘Doing a line,’ refers to a couple who are going out together i.e. my reckoning of it is that back in the day it was considered proper procedure for the matchmaker to always supervise a couple who had just met and the first stage of their courtship would always involve ‘walking out together.’ This ritual of courting someone would begin after a suitable introduction was performed by the matchmaker.

Once the couple had met under the matchmaker’s watchful eye of course then it was expected that they would then go for a nice stroll together down a green country lane. At this stage they were always closely followed by the matchmaker himself. The matchmaker for those of you not familiar with one was usually a local man living in the area whose job it was to introduce eligible young girls to what he deemed to be suitable men for a fee on their marriage. He did this of course for a fee and a big part of his job was negotiating with the potential bride and groom’s families as to what kind of a fortune the young woman in question would be bringing with her into her marriage. This could and often did include any manner of livestock, money or other possessions.

Excellent use of the expression here under "Dry Shite" in a list of Greatest Irish Insults.