Prices are not changing when toggling
For the tax toggle to work properly, you should make sure your tax settings are set correctly in WooCommerce:
1) Tax Rates and Calculations must be switched on
WP Admin > WooCommerce > Settings > General "Enable tax rates and calculations"
2) Tax Rate must be set
WP Admin > WooCommerce > Settings > Tax > Standard Rates + Insert Row with your Tax info. You should set up at least one default tax rate to cover all visitors to your website using empty values (or asterix) for country code, state code, postcode and zip blank so it is applied to all visitors. Unless you ask your visitors location before showing them prices, this should be your default tax rate.
3) Individual Product must be marked as Taxable
WP Admin > Products > Product > Product Data > Tax Status > Product should be "Taxable"
4) Set your default customer location (this is how the tax is worked out if user is not logged in)
WP Admin > WooCommerce > Settings > General > Default customer location should be "Shop base address" and your shop base address should be filled in
5) Optional: Calculate tax based on "Shop Base Address" (Unless you make all your customers register with their address before browsing or your tax system is not based on your store location)
WP Admin > WooCommerce > Settings > Tax > Calculate tax based on "Shop Base Address"