"On tomorrow" vs. "by tomorrow"

You could say one of two things:

I will transfer the amount tomorrow.

I will transfer the amount by tomorrow.

The first indicates that the transfer will occur tomorrow exactly. The second indicates that the transfer might occur before tomorrow, but will not occur later than tomorrow.

This is just incorrect:

I will transfer the amount on tomorrow.

You never use the preposition on to govern adverbs such as today or tomorrow.

To add to JSBangs's answer, the use of "tomorrow" may be masking your real question, which seems to be when to use "on" versus "by". Consider instead:

I will transfer the money on Friday.
I will transfer the money by Friday.

The first says that you will take this action specifically on Friday; the second says that by Friday you will have done it, maybe on Friday or maybe on Thursday or whenever.

I never heard "on tomorrow" before I moved to North Carolina from Pennsylvania. Here, the same person who says "on tomorrow" is equally likely to pronounce "asked" as if it were spelled "axed."

As regards "by tomorrow" I have observed the following distinction:

• If I say "Please transfer your money to me by tomorrow", I mean before the end of business today.
• If I say "I will transfer my money to you by tomorrow", I mean I will transfer the money sometime before midnight tomorrow night. Or before sometime in the morning day after tomorrow.

This is both humorous and true. And it applies equally to the way promises are made to me.

The meaning of "by tomorrow" is in part a function of who will be doing it and partly a reflection of who is the beneficiary. Auto mechanic says "I'll have it done by Friday." If I actually show up to get my car at 7:30 Friday morning, I would not be at all surprised to hear him say "I said 'by Friday,' not 'by first thing Friday.'"