I heard this expression from 7!!(Seven Oops)'s song "Orange". I don't understand what it mean by 涙を預けて.
It's not an idiomatic phrase at all, and it's hard to guess, really. But judging from the whole lyrics, I guess the phrase basically says "I won't cry over this separation any more after this sunset (＝"オレンジ" here)".
Aside from the primary meaning (to deposit something so that one can withdraw afterwards), 預ける also means "to let others decide completely", "to delegate", "to leave something up to someone", etc.
- 物事の処理を人にゆだねる。紛争や勝負の決着を第三者に一任する。「勝負を―・ける」 (デジタル大辞泉)
So I think 「沈む夕焼け オレンジに染まる街に涙を預ける」 can be understood as something like "My tears will go away with this sunset" or "I'll leave my tears in this orange town forever". Obviously he doesn't want to "take back" his tears.
That's how I felt, but I would like to hear opinions from others.