I cannot tell you exactly what file it is because it varies depending on how your portal and its skin were setup. But in general, first, determine if your site is running on the default portal (hopefully not) or if it is a separate portal. This will determine which directory you need to look into for the file in question. If it is a separate portal, go to the root of your DNN install, you will see a directory called “Portals”. Under it, you should see at least the directories "_default" and "0". If you have a separate portal, or multiple of them actually, there will be other directories named with just a number, their skins will be under a directory called "Skins", but I cannot tell you which one your portal is going to be. If there is only one portal created in addition to the default, then its directory name should be "1". Now, under the Skins directory you will have one directory for each skin created and installed for the site, it may be just one there. Do you know the name of the skin you are using??? The directory you are looking for has the name of the skin and usually, the file you are looking for has the name of the skin with the ASCX extension. When you remove the token I mentioned before and save the file, the change will take effect right away.
Edited to add: Depending on how your skin was created, via an HTML file or directly with an ASCX file, you may also want to make the same change in the HTML file if you have one. Again, you may or may not have this file, it would be the same name as the skin and the ASCX but with the HTML (or HTM) extension. In the HTML file you would want to remove the [SEARCH] token. What happens is that the HTML file is processed and the file ASCX is created from it. If you don't make the change in the HTML file and it is reprocessed the Search box may reappear later in the future.
As usual it is recommended that you make a full backup of the DNN directory before making any changes just to be safe.
Again, when in doubt, refer to the skinning documentation, all the stuff above is explained there.
Carlos