You can hide elements using CSS once you have found what selectors to use:
http://en.support.wordpress.com/custom-design/how-to-find-your-themes-css/
To hide tags on posts in the Pinboard theme, add this to your Appearance > Customize > CSS editor:
.entry-meta .cat-links,
.entry-meta .sep,
.entry-meta .tag-links {
display: none;
}
If you also want to hide category and tags inks to indiviual posts, add this instead, which has more specific CSS seletors:
#page .entry-meta .cat-links,
#page .entry-meta .sep,
#page .entry-meta .tag-links {
display: none;
}