details depend on your theme and on the locations where you want to remove the date: i.e. single posts? or archive? or index?
possibility 1:
if your theme uses body_class() and post_class() and has a specific css class for the post date, you can use css to hide the date;
example from Twenty Eleven:
.category-somecat .entry-meta .entry-date { display: none; }
https://codex.www.ads-software.com/Function_Reference/body_class
https://codex.www.ads-software.com/Function_Reference/post_class
possibility 2:
add a conditional statement around the post date output, in index.php, single.php, archive.php, category.php, …
example:
<?php if( !is_category(array('cat1','cat2')) && !( is_single() && in_category(array('cat1','cat2')) ) ) : ?>
existing post date code here
<?php endif; ?>
https://codex.www.ads-software.com/Conditional_Tags
possibility 3:
create category templates for those two categories, starting with a copy of category.php or achive.php of your theme; then remove the post date code from the loop.
https://codex.www.ads-software.com/Category_Templates
