The intransitive verb 届く (to reach) and the transitive verb 届ける (to convey, to deliver) are usually used with tangible objects such as letters. But it's also frequently used with words representing feelings.
The second example is the title of a manga, and people can easily guess that this title actually means "let my love reach you" or something.
Now, 涙を届けて (literally "please deliver tears") is not an established idiomatic phrase. Obviously it's a metaphor, and the actual meaning could be ambiguous. It probably means something like "please notice/feel my sorrow". Those て-form verbs express requests. The following 本当のこと (truth) is grammatically not strongly connected with 涙を届けて, so it would be enough to interpret it as "that (my sorrow) is the truth".
想いを届けて 言えないこと / 言葉にできない (literally "please deliver my feelings, something that can't be said, can't be worded"): So he wants to share his feelings with someone, but he has difficulty in doing so.
明日を届けて (literally "please deliver a tomorrow") is even more ambiguous and poetic, but under this context I think it means "give me a tomorrow" or "may tomorrow come".