What's the difference between Я больше не люблю тебя - I don't love you anymore
я уже не люблю тебя- I don't love you anymore.
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Basically they mean the same thing. However Я больше не люблю тебя is often used when one person stopped loving another person as a consequence of some circumstances or actions that happened immediately or at some point before the speech.
Ты изменил мне и поэтому я больше не люблю тебя - You cheated on me and so I don't love you anymore.
People would unlikely say Ты изменил мне и поэтому я уже не люблю тебя. So people use the больше не люблю тебя phrase when they are reacting to something that made them stop loving, while they may use я уже не люблю тебя, when love is gone but nothing particular happened to move lovers apart.
"Больше не" means "not anymore", "not any longer"
"Уже не" means "[my] need is past due"
"больше не"/"уже не" are both frequently used with "хотеть" (want), but it's almost always "больше не" with "любить" (love).