Because of my immigration status in the US, I have to paper-file my tax form 1040NR-EZ. Then I lost my USPS tracking number and IRS's Where's My Refund doesn't have my info. Should I file an extension before April 15? Is there any drawback from filing an extension if IRS has already received my tax form but not processed it yet?
There is supposedly an online service offered by the IRS that will tell you if they have received your form. In not 100% sure it works with paper filings, but that's what I'd suggest investigating if you need reassurance. There's an IRS-to go application, or a "where's my refund" web tool. See https://www.irs.gov/Individuals
Any paper returns postmarked on or before the official filing date are not considered late. That being said, it's common practice to send returns via registered mail with some form of delivery confirmation. This is so that, should some IRS technician accidentally drop your return in the shredder (or any other unforeseen mishap), that you have some evidence in your defense when they come after you. Most of the time though (in 99.9999% of occurrences), such steps are not needed because everything shows up in the official records, even if it seems to move at a glacial pace.
If I were in your situation, I'd set a calendar reminder to log in on April 15th to check for "received" or "accepted" status, and then not think about it again until that date. If the return is not shown online on that date, then you still have time to call the IRS, file an extension, and even start going about recreating your paper documents.
If IRS received your paper return, but they haven't processed it yet, there is no confirmation.
With IRS understaffed, it can take a while to process paper returns. So, to be on the safe side, you might want to file for extension.
You should be able to do it online. Check here: https://www.irs.gov/filing/extension-of-time-to-file-your-tax-return
Filing for extension won't cause any problems - if IRS receives your paper return on time, they will ignore extension request.