Started getting warning message following host’s PHP upgrade

The issue is this line:

if(function_exists(register_setting))

register_setting is not in quotes. Values without quotes are how you refer to constants in PHP, but there’s no constant with that name, because it’s supposed to be 'register_setting', with quotes.

PHP recognises this mistake, and is interpreting this as if you had included the quotes, but it’s warning you that it won’t do this in the future, so you should fix it now. So just add quotes to fix the issue. You might as well clean up the formatting while you’re at it.

function tl_init(){ 
    if ( function_exists( 'register_setting' ) ) {
        register_setting( 'tl-options', 'tl_logo_src' );
    } 
}

That being said, you can probably remove that condition. That is a backwards compatibility measure to ensure compatibility with older versions of WordPress that do not have this function, but that function has now been in WordPress for 12 years. You don’t need to check it anymore:

function tl_init(){ 
    register_setting( 'tl-options', 'tl_logo_src' );
}