Quoting ♥ inÍs:" I said "start" to fall in love.."
Love is love, no matter at what stage.
I think, personally, that if that scenario happened to me, and especially if that person had feelings for me as well, why NOT keep loving (or fall in love) with them? They have mere weeks to live, I would never deny someone such wonderful feelings like love to someone who doesn't have much time left. I can deal with heartache, someone who is dying deserves to know they are and be loved up until the second they pass, and after.