Ability to modify views tab labels dynamically

This is not possible with standard Views.

It *might* and I stress *might* be possible to do this utilizing hook_menu_alter() (and your module weight must be higher than that of Views, which is 10) and changing the title of the menu item to go to a callback function. You could then run the view to get the number to display.

Please note that doing this could add a significant performance burden to your site.

Thanks! This custom module code works! However it breaks cron, login, save node, flush cache and all other operations (except browsing from page to page) on my website. In the log I find the following error:
Request exceeded the limit of 10 internal redirects due to probable configuration error. Use 'LimitInternalRecursion' to increase the limit if necessary. Use 'LogLevel debug' to get a backtrace.

And it still continues breaking cron, login etc. after I deleted the function TAB_COUNTER_title_callback() or even replaced with a static title: $items['node/%views_arg/questions']['title'] = 'Questions';

I tried to restrict this custom module to work on certain paths only like node/%, but with IF statements in the code it stops working at all:

function TAB_COUNTER_menu_alter(&$items) {
   if (arg(0) == 'node' && is_numeric(arg(1))) {            <-- IF statement turns the whole code non-working.
	$items['node/%views_arg/questions']['title callback'] = 'TAB_COUNTER_title_callback';
   }
}
function TAB_COUNTER_title_callback() {
  $count = db_result(db_query("SELECT COUNT(nid) FROM {node_comments} WHERE nid = %d", arg(1)));
  return 'Questions ('.$count.')';
}

Does anybody have any idea how to make this custom module lighter for the website or restrict to work on certain paths only?

   if (arg(0) == 'node' && is_numeric(arg(1))) {            <-- IF statement turns the whole code non-working.

You have to know that hook_menu_alter is not executed on every page but only on clearing the cache which can happen anytime.
You simply do not need this if, you alter the menu definition, that's it.

Thanks, dereine, for clarification! But still this code (even without IF()) breaks cron, flush cache and save (in any submission forms) actions. In fact, after I refresh the resulting white screen, I see that all these actions have actually been completed "successfully". Besides the white screen I also get the same error in the log:
Request exceeded the limit of 10 internal redirects due to probable configuration error. Use 'LimitInternalRecursion' to increase the limit if necessary. Use 'LogLevel debug' to get a backtrace.
I also tried increasing memory_limit in php.ini and and disabling other modules that might consume memory, but every time I discover that only this simple custom module code produces the error.

What is your current code you use?

I created a custom module tab_counter.module, and everything this file contains is:

function TAB_COUNTER_menu_alter(&$items) {
    $items['node/%views_arg/questions']['title callback'] = 'TAB_COUNTER_title_callback';
   }
}
function TAB_COUNTER_title_callback() {
  $count = db_result(db_query("SELECT COUNT(nid) FROM {node_comments} WHERE nid = %d", arg(1)));
  return 'Questions ('.$count.')';
}

But the same error happens even if the tab_counter.module contains only:

function TAB_COUNTER_menu_alter(&$items) {
    $items['node/%views_arg/questions']['title'] = 'Questions';
   }

So are you sure this path is correct? "node/%views_arg/questions"

Yes, I'm 100% sure, because by using this custom module - on every Node page (e.g. node/1318) I get the existing Views Menu Tab "Questions" successfully renamed to e.g. "Questions (10)", and as I click on the "Questions (10)" Menu Tab I get directed to the right page (e.g. node/1318/questions).

But as I said it breaks cron, login, logout, save, etc. turning into a white screen.

Thank you, dereine, anyway!

I found a totally different solution http://drupal.org/node/802012 with function phptemplate_preprocess() which didn't crash login, logout, save and cron operations on my website.