It seems like the maximum number of seats the US Senate Democratic caucus will obtain is 50. This means that they would be able to pass legislation using the VP to break ties, assuming there is no filibuster. Does the VP also have the ability to break ties for senate rule changes, such as eliminating the filibuster?
Yes. The filibuster is part of Senate rules, and not part of the Constitution. The Senate has a constitutional right to set its own rules, by a simple majority vote.
The procedure is to change the interpretation of the rules (without actually changing the rules which could be subject to a filibustered vote) Someone calls a vote to invoke cloture, and when this vote doesn't achieve the required majority, the presiding officer tells the Senate the vote has failed. Someone challenges that, and tells the presiding officer he's interpreting the rules incorrectly (!). The only way to solve this issue, is by having a simple majority vote on who's right about the rules, which then happens immediately.
The VP has the right to break ties in the Senate, including on votes on rules. (The VP doesn't get to add their vote in situations that require a 60% or 2/3 majority, as those are not ties)
Eliminating the filibuster is called the "nuclear option" which is hyperbole but indicates why it hasn't happened. It might allow you to get legislation passed but it would also allow your opponents to pass legislation that you might want to block in the future. No party has yet had the courage to remove the filibuster on votes that create new laws, for fear of doing long term damage to their future ability to hold the majority to account.
To get an actual change of the rules, filibustering would block attempts to remove the filibuster. There are tricks to get around that, but they appear to require a majority.
From this Brookings Institution article article giving more detail on the filibuster and how it could be changed/ended:
To directly end it, they would need to change the text of the Senate rules... but it actually takes a larger majority (two-thirds) to stop a debate on a rule change. So the minority party would basically filibuster to prevent the rule changes on the filibuster. Because two-thirds is 66.7, there can't be a tie, and the vice president wouldn't vote.
However there is also the perverted workaround of claiming Senate rules aren't being properly followed or unconstitutional. Then it only takes a simple majority vote to form a new precedent to doing things. This challenge is called a point of order. As the Senate Procedure says:
A point of order is not debatable unless it has been submitted by the Presiding Officer to the Senate.
This is the so-called nuclear option. The same method was used in the past to make specific changes to the filibuster rules when passing nominees (first by Democrats for non-Supreme votes, then Republicans for Supreme Court votes) and budgets.
When it comes to the rules of determining simple majority, though, the Constitution says
The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided.
And I could find no other place in the consistution or Senate Procedure that would exclude points of order votes (or appeals of the rulings of the senate leader) from that clause. It may well be that somewhere else in rules of order is buried a key that would keep procedural questions to just the Senate... but otherwise he theoretically just might get a vote.
However the article lists some political reasons why it may be harmful to end the filibuster, such as the inability to place any blame on opponents or reduced ability for senators to push for support on projects important to their state.
And most of all, considering the significant split in the American population's political beliefs, any move to entirely end the filibuster might be short-sighted. Had Democrats eliminated it under Obama when they lost the 60 votes to reach cloture, they would have been stuck without any recourse to hinder legislation in the first two years of the Trump presidency. The filibuster in a sense personifies the significant challenge of finding widespread agreement in America as a whole on many issues, and may be the only hope to forcing parties to find middle ground.