The only way to clear Structured Data errors in Google Webmaster Tools is to either remove the structured data markup so that Google will not attempt to read structured data from the page, or fix the errors in the structured data.
I'll assume that you want to go with route #2, otherwise you wouldn't have added the structured data markup in the first place.
The best process to fix the error will be to click on a structured data type with errors in the Structured Data tab. There you should have your pages with errors listed at the top. Click on one of those URLs, and a window with all of the structured data Google was able to read will pop up. In the case of URLs with errors, it will also show you the reason why your markup is invalid (usually missing some required piece of data). From there you can debug the issue on your page/pages.
When you think you've got the problem fixed you can click the "Test Live Data" button, or use the testing tool here to test the page you believe to be fixed. It will display warnings and errors, but only errors will prevent Google from using your structured data, warnings will not.
Once the errors on the page are fixed, they will take some time to be reflected as so in Google Webmaster Tools, as the pages will need to be crawled by Google again. This can take weeks sometimes depending on the popularity of the pages.