Note: This is a historical reference, not a current security baseline. Magento 1 is end-of-life.

It is not possible to remove products from cart or minicart after installing SUPEE-11219 or upgrade to Magento 1.9.4.3. There are no any errors shown.

Cause

Most likely cause of this problem are missing form keys in custom design theme used in template/checkout/cart/sidebar/default.phtml. As seen in corresponding changes for RWD theme:
--- app/design/frontend/rwd/default/template/checkout/cart/sidebar/default.phtml
+++ app/design/frontend/rwd/default/template/checkout/cart/sidebar/default.phtml
@@ -36,7 +36,14 @@

        __('Remove This Item') ?>
+        
+            __('Remove This Item') ?>
+        

Solution

Custom theme should be updated accordingly after SUPEE-11219 patch to allow form submission with form keys as shown in changes for RWD theme.