tappa is also used for abilities such as minnet and talförmågan ('the ability to speech'). förlorar isn't wrong either but slightly higher style.
If you tappar pengar, it only means that you drop it, if you förlorar pengar it means that you lose it, either in a game or like an unsuccessful venture.
We generally prefer to speak about our own body parts (and similar things) as determinate rather than with possessive pronouns. In this case, there is the added complication that tappa minnet is like a set combination. So while it isn't really grammatically wrong to say Har du tappat ditt minne?, it sounds really really awkward. It also makes it sound like you're speaking about 1 specific memory, whereas the normal meaning of tappa minnet is 'lose one's memory' as in losing the ability to remember things. Therefore it's more OK to say tappat ditt minne about an usb stick than about what's in your head.
It means "your memory" in both Swedish and English. Things closely associated with or connected to the subject are often given in definite form.
Har du tappat plånboken? (Have you lost your wallet?) It's understood to mean "your wallet" and not merely "the wallet" which could belong to anyone.