This looks to be quite a challenge. In general a media print css file will take care of this for a given webpage. I assume this will be published as a web form. If you are also using XMLFormView, you can include the style in your aspx page or another web part on that page.
Something like this.
<link rel="stylesheet" media="print" type="text/css" href="http://WebServerorSharePontServer/Docs/print.css" />
inside the print.css I wold try something similar to
input{border:none;}
Note. I haven't tried this so best of luck.