In my experience, other tracks benefit a little for a short time when another track in their area closes. Overall the whole area's racing suffers.
As Andy says, some teams will try to keep racing so they will go to other tracks. But a lot of them will cut back or quit because of the added travel costs and / or costs to change their equipment to make it legal and competitive at those other tracks. It has just become harder to finish in the money, and for teams not used to traveling there will be a steep learning curve. If you have to change rules packages (to or from NASCAR, for example) that can be really expensive - that's mainly how Caraway killed itself off. Sponsorship can get harder if teams' current sponsors aren't interested in staying if the new track is out of their market reach. Those sponsors might not go away immediately, but if they're not getting acceptable ROI then they probably won't renew.
There can be a ripple effect. Fewer tracks yield fewer racing opportunities, which can force some teams to quit while others decide to switch to another league. For example, in central NC back in the 1980s / 1990s we had an unofficial dirt late model circuit of Ace on Fridays, 311 on Saturdays, and a track in Wilson (sorry, don't remember its name) on Sundays. When the track in Wilson closed, we saw a few less teams since they moved to where they could earn enough money to keep going. Then Ace paved, so dirt cars were useless there. 311 by itself, and its inability to cooperate with Friendship, was not enough to keep the late model teams around - some shut down, some tried other series. Although there were other factors too, dirt late model racing in central NC is now a shadow of what it had been.