Does fixing PHP notices improves website performance?

It’s worth doing so, though performance isn’t always going to improve, notices are usually a sign of broken-ness or bad code.

In the example you cited, there’s actually a mistake made and a potential security issue:

$geolocation = esc_sql($_COOKIE['geolocation']);

• the code assumes there’s cookies, and that a geolocation cookie is present
• it uses esc_sql which should not be used here, esc_sql is almost never used by WP developers as it’s mainly found inside the WPDB class
• a Geolocation is not an SQL statement
• esc_sql is an escaping function, not a sanitising function!
• there is no default value for when it’s undefined

A better way of doing this would be:

$geolocation = '';
if ( !empty( $_COOKIE['geolocation'] ) ) {
    $geolocation = wp_strip_all_tags( $_COOKIE['geolocation'] );
}

Here I’m not sure what the default value should be, or the format of this value, but you would swap out wp_strip_all_tags for an equivalent that sanitises it. You would also want a validation step to ensure it actually is a geolocation.

Notices and warnings can be caused by lots of things, and a lot of them may have no impact on performance, but they can be signs of bugs, security problems, or just awful code quality! If you buy a product and it starts spamming you with PHP notices then that’s a bad sign.

PHP warnings and notices are like mugs with designs that peel off in the dishwasher, door handles that aren’t screwed on properly, clothes in a shop that have stains on them, or cars that start making weird and ominous noises. Just because it doesn’t make your site faster doesn’t mean they shouldn’t be fixed.