Beta iOS/stable watchOS conflicts? I have an iPhone that I fully plan to put on the iOS 11 public beta later this month, but I also have an Apple Watch that's going to be stuck on watchOS 3. Are there going to be any issues from that, such as my phone expecting my Watch to be on watchOS 4?

(I know that it's impossible to predict what bugs are going to be in a beta, I'm asking more about if anyone has experience running an iOS public beta alongside an Apple Watch on the latest stable watchOS release.)
The best indication that this is okay lies in the fact that iOS betas are generally available to the public, while there is no public beta program for watchOS. So yes, it’s fine to run an iOS beta with a stable release of watchOS.
Apple has no mechanism in place to force you to run a certain version of watchOS with a given version of iOS (other than the minimum version required to install watchOS). They don't force you to update watchOS along with iOS. Barring such guarantees, they cannot make any assumptions as to the version of the operating system you’re running on a separate product.
However, as with any beta software always be prepared for unexpected bugs. For instance the iOS 11 Beta 2 release notes mention the following issues with earlier versions of watchOS:

Known Issues
Pairing an iPhone running a beta of iOS 11 and an Apple Watch running a version of watchOS earlier than 4.0 may fail. (32587160)
Workaround: Attempt to pair the Apple Watch again.
• In the Watch app, changes to Mail &gt; Include Mail are not saved. (32553544)
Resolved Issues
• On an Apple Watch running a version of watchOS earlier than 4.0, location activity now correctly sets the location arrow in the status bar or in Settings. (31496053)
• watchOS email alerts now use the configured settings. (32332614)