When I TRY to remote into my other computer, and the other computer is left logged in, the remote computer gives the message:

 "is trying to connect to this computer. If you allow, you will be disconnected, but you can resume later. Do you want to allow this connection?".

At this point I must physically be at the computer to click "yes" and allow the connection. Is there any way to disable this option so that I can just connect without the remote computer having to choose yes?
Yes, maybe there is a way to do that. But it makes a difference to what you're trying to. Is this a case where you're going to be away from your computer, some distance, and you want to LOG in to your computer with CONTROL of the windows desktop?
Question, but program are you using? If you are using a Windows remote assistant , it requires two people. One person giving t the assistance and the other is the user who requires some help.
Now I may be WRONG, but I think you have to use a different program.

What versions of Windows are on the two computers?  This may help: http://windows.microsoft.com/en-US/windows-vista/Remote-Desktop-Connection-frequently-asked-questionsI was using my laptop (vista) and trying to connect with "remote desktop connection" to my desktop XP computer on my home network. If another user was logged into the xp machine and I was trying to remote into it I would have to go to the computer and click yes on the prompt: 

"is trying to connect to this computer. If you allow, you will be disconnected, but you can resume later. Do you want to allow this connection?".

The solution is to disable user fast switching, then remoting into the xp desktop via remote desktop connection will log off the other user.You should try TeamViewerbro try to team viwer