Text form field - Word 2010

When you say "shadow", are you referring to the grey shading that fills the form field? If so, you can turn that off by clicking the Form Field Shading button on the Legacy Forms gallery, as shown here:

It's a toggle button, that is, it turns the shading off with one click and on again with the next click.

If you turn off the shading, the only thing that indicates the presence of a form field is the pair of grey square brackets that represent the field's built-in bookmark. You can turn that off, too -- but only on your computer -- by going to File > Options > Advanced > Show document content, and turning off the option "Show bookmarks".

If instead you mean that the text itself displays a shadow, like the first line below:

then you must turn off the forms protection and select the form field; go to the Home tab and click the Text Effects button; choose Shadow and then click No Shadow in the secondary menu; and finally reprotect the form.