Ytukyg

If necessary, Congress can pass a spending bill without the president's support.

Currently Trump claims he will veto any bill which doesn't include funding for the wall. However if two-thirds of each chamber of Congress agree, they can override a presidential veto and end the shutdown.

The current Senate has already passed a spending bill without funding for Trump's wall once with a veto-proof majority of 100-0. Instead of voting on that bill though, the Republican-lead House of Representatives voted on a different bill with wall funding that was sure to fail in the Senate.

The Senate won't change much in the next Congress, but the House of Representatives will have new leadership and can try to pass a similar bill to what has previously passed the Senate. If the Senate votes similarly and the House gets enough votes they can end the shutdown with or without Trump.

The government shutdown is caused by a lapse in funding for government operations and agencies.

The Antideficiency Act mandates that the government cannot incur "obligations or the making of expenditures (outlays) in excess of amounts available in appropriations or funds". As explained by The Hill, this act "provides the framework for which government functions temporarily cease and how employees are impacted".

Thus, the only way to end a government shutdown is to pass appropriations legislation. This can be in the form of a regular appropriations bill (a yearly budget) or a continuing resolution (CR; a short-term funding bill). Without any appropriations legislation passed, the shutdown will continue indefinitely.

Currently, both parties are looking to pass a continuing resolution. However, since there was no agreement on the amount of funding to provide for border security, the Senate passed a CR which does not include funding for border security while the House passed a CR which includes $5.7 billion for border security. Since there was no agreement between the Senate and the House, the government shutdown continues to date.

If Congress - House and Senate - cannot agree to a bill with 2/3 majority, two years go by, and there will be new elections for all House seats (and about a third of the Senate). Some voters will be annoyed with the shutdown, and vote for someone who is more interested in ending it.

If that is not sufficient, two more years will go by, and more people will be annoyed. And so forth, until enough representants are elected that can agree on ending it.

There is no limit to the time, though; theoretically, this could go on for centuries.

Can it last forever?

The shutdown continues until a bill passes both houses and is either enactes by the president or passed through the veto override procedure. Theoretically, the stalemate could never be resolved, and the government might fail or something, or maybe 2/3 of the state legislatures call for a constitutional convention in order to try to fix the problems. None of that seems terribly likely given the political incentives, but in the American system, there is no higher political power than Congress and the president under the Constitution.

Practically speaking, though...

At some point, one side or the other will give in. This is a game of chicken, just in the halls of Congress instead of the school playground. The entire House will be elected again in two years, so the probability of a shutdown lasting beyond that is essentially 0. Each party knows this, and so they each want to end the shutdown before it becomes politically precarious.

The key problems right now are that the House and Senate are run by opposing parties and that the Senate must invoke cloture on any funding bill, i.e. the Senate must have at least 60 votes to end debate.

Speculation

If I were to make a guess, I'd say that invoking cloture on a bill funding a wall at $5B is the biggest hurdle, since it would involve seven Senate Democrats defecting. Maybe Mitch McConnell could convince a few of the more centrist or conservative Democrats like Jon Tester (D-MT) or Democrats from "red" states like freshman Kyrsten Sinema (D-AZ) to support a bill with wall funding. Absent a deal between Trump and Schumer (D-NY), the Senate is going to be a hard sell.

I think the House may actually be a bit easier, since the Republicans would need only to make a deal with the House leadership to bring a wall-funding bill to the floor and then get at least 18 Democrats to support it.

In percentage terms, only 8% of House Democrats would need to defect to pass the wall funding as opposed to 14% of Senate Democrats.

I would be surprised if it could go on for more than another two years. The classification of essential vs. nonessential is rather arbitrary. I'm pretty sure that the "nonessential" personnel who have to work without getting paid (because they are, in fact, necessary) are going to win their lawsuit that they can't be required to work without getting paid. (For the record, their paychecks are "deferred". Their living expenses are not.) I note that all the reference attorneys in the federal judicial system are essential and are being paid, so the essential processes of Americans suing each other can go on unchecked.

Nonetheless as people discover which "nonessential services" are in fact essential for their lives (income tax refunds, perhaps?), they will grow more and more disenchanted with the shutdown. Most services (all perhaps?) are there because someone wanted them. Enough to want them to continue? Certainly in the aggregate, and they may be willing to allow the whole mess to continue and have Congress decide which programs to kill by a less contentious method. (And see how well that has worked. Farm subsidies are mostly gone -- except to the ginormous agribusinesses. [Well, the subsidy line item in the federal budget is just as big, but small farmers complain that they aren't getting subsidies any more. Either they're lying, all of them, or the payments are all going to the big corporations.])

So here's my prediction. A compromise will be reached. A wall will not be built. Some other bone will be sent Trump's way so that he can claim "victory". Fox News will go on and on about how he got the best deal possible in the face of Democratic intransigence, because of what a great negotiator he is. All the work will be done in the House and Senate, but on Fox News Trump will get the credit and Pelosi the blame, and in the NY Times and the Washington Post the reverse. CNN will be too scared to take a stand, except for Rachel Maddow. This post will be banned for being too overtly excessively political.