Calculate difference between dates over multiple iterations

It seems like you are looking for: CONTAINSSTRING( 'Table'[Status], "request" )

I think you can just incorporate that into your current formula to get what you want.

You may want to consider incorporating variables rather than use EARLIER to make your DAX a little more readable. Eg

Duration (min) = 
IF( 
    CONTAINSSTRING( 'Table'[Status] , "approval" ), //ignore unless we are in approval row
    VAR _thisTime = 'Table'[Time Status was Set] 
    VAR _thisRec = 'Table'[Record Number]
    VAR _prevReqs = FILTER( 
        ALL( 'Table'[Record Number], 'Table'[Status], 'Table'[Time Status was Set] ), 
        'Table'[Record Number] = _thisRec
        && 'Table'[Time Status was Set] < _thisTime 
        && CONTAINSSTRING( 'Table'[Status], "request" ) 
    )
    VAR _lastReqTime = CALCULATE( MAX( 'Table'[Time Status was Set] ), _prevReqs, REMOVEFILTERS( 'Table' ) )
    RETURN
    DATEDIFF( _lastReqTime, _thisTime, MINUTE )
)