I paid $22 foreign tax on my 1099-DIV. Do I need to fill out Form 1116?

No, you do not need to file a Form 1116 if foreign taxes is not more than $300.  After you enter info under 1099-DIV section, you should see the $22 on Schedule 3 line 1 and Form 1040 line 20.

To enter 1099-DIV in the program, here are the steps:

• Sign in to your account, select Pick up where you left off
• To the right upper corner, under Select, type in "1099div" then Enter
• Select Jump to 1099div
• Follow prompts 
• On the 1099-DIV screen, make sure to check the box "My form has info in other boxes" to expand the options
• Make sure box 7 Foreign tax paid is filled. 

You only need to file a Form 1116 if foreign taxes paid exceeds $300. 

For more information, click here: https://www.irs.gov/instructions/i1116  ( Under Election To Claim the Foreign Tax Credit Without Filing Form 1116)

Entering $22 in box 7 does not do anything to Schedule 3 line 1. It has '0' inside the box and Form 1040 line 20 is blank.

It means that you were not able to utilize that tax credit this year and will be carried over into the next tax year. This may be because you had no taxable income this year thus no tax liability to apply tha credit to.  

Do you know if there is a way for me to bypass the screen saying I need Form 1116? I'm not able to file because it wants me to upgrade to another version of TT.

Yes go to the 1099-DIV entry in Box 7 and remove the $22 amount.