I created a tab (website) and paste link to a sharepoint site, it works fine in Teams desktop but it doesn't work in Teams web app (opened from a web browser). Is this a known issue and going to be fixed?
tabs link to website other than sharepoint works fine.
Yes, the integration between Teams and SPO when using the Web Client has still some problems Today so you can potentially experience the kind of issues you have described here. Unfortunately there are not workarounds to fix it.