Any system of government can be changed, so in that sense this is an easy question. There are several countries where the Queen was previously Head of State but no longer is--see, for example, South Africa which had a referendum in 1960 to become a republic.

The question in the UK is obviously much more interesting. There are certain properties, such as Sandringham House or Balmoral, which are the private property of the Queen and are passed down in her family as a normal inheritance. These would be completely unaffected by any change in the status of the monarchy. Beyond that, what you are asking about mainly involves the Crown Estate. These are a wide variety of holdings which are not owned by the Queen individually, but by the monarch by right of their title. The functioning of the Crown Estate is governed by the Crown Estate Act 1961 which set up an independent commission to manage the revenues of the estate and transfer the surplus to the treasury. From this surplus, the monarch historically received a grant called the Civil List (until 2011 when this became the Sovereign Grant) to support the Royal Household.

Much like the Duchies of Lancaster and Cornwall, the Crown Estate is managed to provide income for the Treasury and the sovereign. To slightly break the 20 year rule, the 2011 Sovereign Grant Act set a funding formula for the monarch at an indexed 15% of the revenues of the Crown Estate, alongside the revenues of her private estate, the Duchy of Lancaster. In a hypothetical future scenario of republicanism, how to divide the Crown Estate would be a great debate. There is no formula in place dictating how this should be done, but in all likelihood assuming a peaceful referendum took place, the majority of these assets would transfer to Government control.

However, the Queen would certainly not be left broke. She would retain the property that is owned privately as well as a large amount of valuable artwork, furniture, etc. In a sense, if you think of the Civil List/Sovereign Grant as a salary, then the Royal Family has slowly been purchasing valuable items since the Civil List Act 1760 with their own "salary." Further, the Duchy of Lancaster is the Queen's private holding and the Duchy of Cornwall is the Prince of Wales' private holding. This was done intentionally to give the monarch and heir sources of revenue independent from government. Barring expropriation of these properties, the Queen could expect to live quite comfortably; Lancaster posts a budget surplus of around £20mn annually.

That said, this is not really a AskHistorians question. We're dealing with hypotheticals in the present day, not a set event in the past. I can explain how the Duchy of Lancaster came into being, how the Sovereign Grant Act came to be passed, etc, but to actually answer the question of what the financial implications of a republican Britain would be for the monarch is a question for economists.