You can keep this tape until tomorrow.

Literal

Tomorrow [until-まで] this tape [object-を] have-continuing good [assertion-よ].

持ってていいよ is contracted from 持っていていいよ — ~ていい ('it's okay to') gives permission. The double て-form contraction (持って + いて → 持ってて) is common in casual speech. よ adds assertion.