Who pays for travel for a conference paper completed at previous job?
First of all, I would recommend that one never submit a paper to a conference unless you have some idea who will be going to present it and how their travel will be supported.  I would thus have advised you to discuss the possibility of travel funding with your current advisor before you submitted.  Of course, it is too late for that now, but it may not be for others who read this answer.

That said, depending on the particulars of arrangements, it could easily be either institution that pays for travel.  I have often known people to have one last trip paid for by their former advisor, since it is the completion of their last piece of work.  More often, however, I have known the new employer to foot the bill, as presenting at conferences is generally part of their work as a researcher---it's worth noting that the new employer will be paying for the trip in terms of your salary, even if they don't cover the travel expenses.  I have also at some points known people to end up having to pay for their own travel on a junction like this, but that is unusual and can indicate trouble building with the new employer.

Bottom line: it could be any arrangement.  The old employer may be willing, especially if you arrange before you leave. The new employer is more likely to be willing, if they are truly interested in your training as a postdoc and not just exploiting you for cheap labor.