To remove the Estimate shipping and taxes block you’ll need to customize the layout of your theme.
In order to do this you can copy app/design/frontend/base/default/layout/checkout.xml intoapp/design/frontend/default/grayscale/layout/ (you need to replace default/grayscale with the path to your theme files) and after that customizeapp/design/frontend/default/grayscale/layout/checkout.xml removing the following line there:
Another, more correct way (as it will not override whole default layout, so will be more upgrade-friendly) would be to edit local.xml file in app/design/frontend/default/grayscale/layout/ and add the following content right above the closing at the bottom:
<block type="checkout/cart_shipping" name="checkout.cart.shipping" as="shipping" template="checkout/cart/shipping.phtml"/>
After that you need to refresh Layouts cache at System > Cache Management
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