While the answer is "No," the repercussions of using laser paper in an inkjet printer aren't as bad as using inkjet paper in a laser printer. Using inkjet paper in a laser printer can severely damage your printer, though there won't be too much printer damage done to your inkjet printer if you use laser paper in it.
Just like the inkjet paper has a coating, there's also a coating on laser paper that doesn't play well with inkjet printers. No, the coating won't mess up your printer like the inkjet paper coating will mess up your laser printer, but it will have an impact on your print quality.
The ink from an inkjet printer doesn't stick to paper coated for laser printers. Your document will easily smear or blot when it comes out of the printer. Paper coated for laser printers is not meant to absorb ink like paper coated for inkjet printers; it's meant to have toner set and melted on top of it.
Smearing is especially evident with glossy paper that's meant for laser printers but used in inkjet printers. Glossy papers are more heavily coated than matte papers so it only makes sense that the smearing be more evident on glossy paper than matte paper.
So, using laser paper in an inkjet printer is not as bad as using inkjet paper in a laser printer, but still not recommended. If you want to get great, high quality prints from your printer, make sure you're using paper that's made for your type of printer (whether it's inkjet or laser).
Check out some postcard paper for inkjet, laser, and both inkjet and laser printers!