In order to remove the shopping cart icon in the main menu, you can follow the steps below:

Step 1: 

You can simply deactivate the WooCommerce plugin and the icon won't be visible anymore. Please go to Dashboard > Installed plugins: 

Step 2: 

In case you are going to use the shop functionality, then please add the code below to the Custom CSS area:

.fixed-header-box .cart-dropdown {
display: none;

You can also hide the shopping cart icon on lower resolutions (mobile view) by adding the custom CSS code below:

@media (max-width: 959px) { .responsive-layout header.main-header .logo-wrapper .mobile-logo-additions .vamtam-cart-dropdown-link {  display: none !important; }}

Here's a guide: