Certain browsers will always search for a favicon and will carry on doing so until you place the file. There is absolutely no reason why you should need to put a 307 or any other redirect method for that matter simply to get rid of a 404 error.
404's does not imply your site is broken
It's important to note that errors 404 does not indicate that something is broken on the site, if a browser is requesting a file that simply doesn't exist on your server then a 404 response is completely valid.
Redirecting a Favicon 404
Redirecting a favicon with 307 is a bad idea, browsers will still request the file and by adding a 307 you are adding an additional request to your server therefor adding delay unnecessary on the page.
You can design a favicon even though the rest of the site uses no graphics doesn't mean you can't have a favicon. It's actually rather good for tab bars in modern browsers.
Upload a 0 byte favicon.ico to your server, but that means the default browser favicon will not be displayed
Best method - Remove these errors from your logs, you can filter out this information from your logs and that way you never see them... Remember browsers will always request this file and even with redirect it doesn't stop the problem - redirecting is fixing nothing but more of a cosmetic fix for your logs, the correcy way is to filter these errors from your logs.. In Apache you can remove these errors by using:
Redirect 404 /favicon.ico
ErrorDocument 404 "No favicon
For alternative server platforms you should find many solutions on Stack Overflow and Google.