Go destructors?
In the Go ecosystem, there exists a ubiquitous idiom for dealing with objects which wrap precious (and/or external) resources: a special method designated for freeing that resource, called explicitly — typically via the defer mechanism.
This special method is typically named Close(), and the user of the object has to call it explicitly when they're done with the resource the object represents. The io standard package does even have a special interface, io.Closer, declaring that single method. Objects implementing I/O on various resources such as TCP sockets, UDP endpoints and files all satisfy io.Closer, and are expected to be explicitly Closed after use.
Calling such a cleanup method is typically done via the defer mechanism which guarantees the method will run no matter if some code which executes after resource acquisition will panic() or not.
You might also notice that not having implicit "destructors" quite balances not having implicit "constructors" in Go. This actually has nothing to do with not having "classes" in Go: the language designers just avoid magic as much as practically possible.
Note that Go's approach to this problem might appear to be somewhat low-tech but in fact it's the only workable solution for the runtime featuring garbage-collection. In a language with objects but without GC, say C++, destructing an object is a well-defined operation because an object is destroyed either when it goes out of scope or when delete is called on its memory block. In a runtime with GC, the object will be destroyed at some mostly indeterminate point in the future by the GC scan, and may not be destroyed at all. So if the object wraps some precious resource, that resource might get reclaimed way past the moment in time the last live reference to the enclosing object was lost, and it might even not get reclaimed at all—as has been well explained by @twotwotwo in their respective answer.
Another interesting aspect to consider is that the Go's GC is fully concurrent (with the regular program execution). This means a GC thread which is about to collect a dead object might (and usually will) be not the thread(s) which executed that object's code when it was alive. In turn, this means that if the Go types could have destructors then the programmer would need to make sure whatever code the destructor executes is properly synchronized with the rest of the program—if the object's state affects some data structures external to it. This actually might force the programmer to add such synchronization even if the object does not need it for its normal operation (and most objects fall into such category). And think about what happens of those exernal data strucrures happened to be destroyed before the object's destructor was called (the GC collects dead objects in a non-deterministic way). In other words, it's much easier to control — and to reason about — object destruction when it is explicitly coded into the program's flow: both for specifying when the object has to be destroyed, and for guaranteeing proper ordering of its destruction with regard to destroying of the data structures external to it.
If you're familiar with .NET, it deals with resource cleanup in a way which resembles that of Go quite closely: your objects which wrap some precious resource have to implement the IDisposable interface, and a method, Dispose(), exported by that interface, must be called explicitly when you're done with such an object. C# provides some syntactic sugar for this use case via the using statement which makes the compiler arrange for calling Dispose() on the object when it goes out of the scope declared by the said statement. In Go, you'll typically defer calls to cleanup methods.
One more note of caution. Go wants you to treat errors very seriously (unlike most mainstream programming language with their "just throw an exception and don't give a fsck about what happens due to it elsewhere and what state the program will be in" attitude) and so you might consider checking error returns of at least some calls to cleanup methods.
A good example is instances of the os.File type representing files on a filesystem. The fun stuff is that calling Close() on an open file might fail due to legitimate reasons, and if you were writing to that file this might indicate that not all the data you wrote to that file had actually landed in it on the file system. For an explanation, please read the "Notes" section in the close(2) manual.
In other words, just doing something like
fd, err := os.Open("foo.txt")
defer fd.Close()
is okay for read-only files in the 99.9% of cases, but for files opening for writing, you might want to implement more involved error checking and some strategy for dealing with them (mere reporting, wait-then-retry, ask-then-maybe-retry or whatever).
runtime.SetFinalizer(ptr, finalizerFunc) sets a finalizer--not a destructor but another mechanism to maybe eventually free up resources. Read the documentation there for details, including downsides. They might not run until long after the object is actually unreachable, and they might not run at all if the program exits first. They also postpone freeing memory for another GC cycle.
If you're acquiring some limited resource that doesn't already have a finalizer, and the program would eventually be unable to continue if it kept leaking, you should consider setting a finalizer. It can mitigate leaks. Unreachable files and network connections are already cleaned up by finalizers in the stdlib, so it's only other sorts of resources where custom ones can be useful. The most obvious class is system resources you acquire through syscall or cgo, but I can imagine others.
While they can patch over some mistakes, you shouldn't rely on finalizers. They don't run until GC does. Because the program could exit before next GC, you can't rely on them for things that must be done, like flushing buffered output to the filesystem. Assuming GC does happen, it might not happen soon enough: if a finalizer is responsible for closing network connections, maybe a remote host hits its limit on open connections to you before GC, or your process hits its file-descriptor limit, or you run out of ephemeral ports, or something else. So it's much better to defer and do cleanup right when it's necessary than to use a finalizer and hope it's done soon enough.
You don't see many SetFinalizer calls in everyday Go programming, partly because the most important ones are in the standard library and mostly because of their limited range of applicability in general.
In short, finalizers can help by freeing forgotten resources in long-running programs, but because not much about their behavior is guaranteed, they aren't fit to be your main resource-management mechanism.