Yes, airo also means "to lift up" but that is not the only definition and the best way to know if it means remove or lift up is to consider the context of John 15.
airó: to raise, take up, lift
Original Word: αἴρω
Part of Speech: Verb
Transliteration: airó
Phonetic Spelling: (ah'-ee-ro)
Short Definition: I raise, lift up
Definition: I raise, lift up, take away, remove.
http://biblesuite.com/greek/142.htm
αἴρει (airei) — 8 Occurrences
Matthew 9:16 V-PIA-3S
BIB: ἱματίῳ παλαιῷ αἴρει γὰρ τὸ
NAS: for the patch pulls away from the garment,
KJV: that which is put in to fill it up taketh from
INT: clothing old tears away indeed the
Mark 2:21 V-PIA-3S
BIB: δὲ μή αἴρει τὸ πλήρωμα
NAS: the patch pulls away from it, the new
KJV: that filled it up taketh away from
INT: moreover not tears away the filling up [piece]
Mark 4:15 V-PIA-3S
BIB: Σατανᾶς καὶ αἴρει τὸν λόγον
NAS: comes and takes away the word
KJV: and taketh away the word
INT: Satan and takes away the word
Luke 8:12 V-PIA-3S
BIB: διάβολος καὶ αἴρει τὸν λόγον
NAS: comes and takes away the word
KJV: and taketh away the word
INT: devil and takes away the word
Luke 11:22 V-PIA-3S
BIB: πανοπλίαν αὐτοῦ αἴρει ἐφ' ᾗ
NAS: him and overpowers him, he takes away from him all his armor
KJV: him, he taketh from him
INT: complete armor of him he takes away in which
John 10:18 V-AIA-3S
BIB: οὐδεὶς αἴρει αὐτὴν ἀπ'
NAS: No one has taken it away from Me, but I lay
KJV: No man taketh it from
INT: No one takes it from
John 15:2 V-PIA-3S
BIB: φέρον καρπὸν αἴρει αὐτό καὶ
NAS: fruit, He takes away; and every
KJV: not fruit he taketh away: and every
INT: bearing fruit he takes away it and
John 16:22 V-FIA-3S
BIB: ὑμῶν οὐδεὶς αἴρει ἀφ' ὑμῶν
NAS: and no one [will] take your joy
KJV: joy no man taketh from you.
INT: of you no one takes from you
http://biblesuite.com/greek/airei_142.htm