removeAll(closure) in Swift
Some general remarks:


The method name removeAll is misleading, as it does not remove
all elements from the array. I would call it for example removeMatching.
The method parameter is a closure, but acts as a predicate, so
predicate might be a better name.
The counter variable in the method is actually the array index
so you could call it index.
The method documentation can be improved (details at the end of
this answer).


Your concrete problem about modifying the loop counter inside the
loop can easily be solved by traversing the array indices in 
reverse order:

for var index = count - 1; index &gt;= 0; index-- { ... }


Using stride() allows you to have a constant loop variable: 

for index in stride(from: count - 1, through: 0, by: -1) { ... }


Putting it all together:

extension Array {

    /// Removes all items that satisfy the predicate:
    mutating func removeMatching(predicate: (T) -&gt; Bool) {
        for index in stride(from: count - 1, through: 0, by: -1) {
            if predicate(self[index]) {
                removeAtIndex(index)
            }
        }
    }
}


An alternative implementation would be 

extension Array {

    /// Removes all items that satisfy the predicate:
    mutating func removeMatching(predicate: (T) -&gt; Bool) {
        self = self.filter { !predicate($0) }
    }
}


The first method removes the matching elements, while the second method
creates a new array and then replaces self. Which one is more 
efficient depends how many elements are actually removed in relation
to the array size.

Notice that in Swift 2, Element should be used in place of T.

As mentioned above, the inline method documentation can be improved
(I took this information from the SO thread Swift Documentation Comments). Example:

extension Array {
    /// Removes all items that satisfy the `predicate`
    ///
    /// :param: predicate A boolean predicate.
    mutating func removeMatching(predicate: (T) -&gt; Bool) {
        self = self.filter { !predicate($0) }
    }
}


Now "Option-Click" will show you 





Addendum: A removeAll(where:) was added to the Swift Standard library in Swift 4.2, so there is no need to define your own method for that purpose anymore. 

As explained in SE-0197 this method is supposed to be both safe and efficient.