Some are speculating that President Trump will fight the vote count in court all the way until January 20th, when the next-term President (presumably Biden) is sworn in. But the Electoral College votes on December 14th, and in the end it's the Electoral College that determines the President, not the votes. Wouldn't the vote count in each state essentially be irrelevant after that? I understand why the President is contesting the votes now, but would there be any point in continuing to do so after December 14th?
No, in terms of selecting the next President, each State's popular vote count ceases to be relevant as soon as the State certifies the results and selects their electors which will occur even before the Electoral College actually votes. Of course, there is still a moral victory in a popular vote victory, so there's some benefit there, but it won't affect the election.
The more likely reason to hold this position is to buttress the completely unfounded argument that the election was stolen from Trump due to massive voter fraud. This story is completely made up, and the Trump campaign can't even pull together the most basic level of evidence, but by asserting the statement and "fighting", they can rile up their base. The important date here is not December 14th, but January 5th, when Georgia will have their double Senate runoff election. Getting the base angry will, they hope, enhance turnout and help them hold onto their majority. As Robert Costa, a political reporter for the Washington Post reports on twitter:
Based on my convos with Rs over wknd, most everything McConnell does from here on isn't about January 20th (inauguration day & working with Biden) but January 5th (the Georgia run-off elections). To win the latter, Rs believe the base must be stoked, esp in a fast-changing state.
Another reason, as GeoffreyBrent pointed out, is for fundraising purposes. The Trump Campaign is already sending out fundraising emails soliciting donations for an "election defense fund", but most/all of the money is actually going to pay off election debt or directly into the operating accounts. This is, of course, similar to 2016 when Green party candidate Jill Stein raised more than $7 million for a "recount" campaign, which produced few results.
There is a far-fetched hypothetical possibility that developments between the voting of the Electoral College on Dec 14 and the counting of the votes on Jan 6 could influence whether Congress rejects electoral votes from some particular state (or, in the case where two different purported sets of electors sent votes, it could influence which set of electoral votes Congress accepts).
Of course, to reject electoral votes would require a majority of both houses of Congress voting to reject, and given that the Democrats have a majority in the House, this is highly unlikely.