These lines are from the song: "Like Someone in Love," which was written in 1944 and originally sung by such stars as Bing Crosby and Frank Sinatra. When I hear "limp as a glove" in this context, it reminds me of typical gloves of the period, which could be made of soft materials such as silk, satin, or leather. They are limp when held in the hands and not worn on the hands. Thus, the person in love becomes limp (useless or non-functional, or lacking internal strength) when just looking at the one he or she loves. To limp here does not mean to walk with a limp, but to be limp: "Lacking internal strength or structure; not stiff or firm:" (Oxford Dictionary Online) Of course not all gloves are limp: some winter gloves and boxing gloves included.