Extracting sequences from a list?

Try this one

ReplaceList[expr, {__, PatternSequence["Open",v__ /; Count[{v}, _String] == 0, 
            _String], __} -> {v}]

Here's a plain pattern approach, I'm not quite sure how robust it is:

ReplaceList[expr, {___, "Open", x : Except["Close"] ..., "Close", ___} :> {x}]

Also take a look at Longest and Shortest, which may come in handy.

In versions 10.1+, you can use SequenceCases:

 lst={1, 2, 3, "Open", 3, 2, "Close", 9, 3, 4, "Open", 1, 0, "Close", 3, 5};
 SequenceCases[lst, {"Open", x:Except["Close"].., "Close"} :> {x}]

{{3, 2}, {1, 0}}

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