Example:

Here's how you can remove the logo.
First, you will need a FTP program to download your index.template.php file with. The file can be found in the <theme_name> directory inside of the Themes directory.
Than Locate and delete following code:
For SMF1.1.x (Core)
<td align="right" class="catbg">
<img src="', $settings['images_url'], '/smflogo.gif" style="margin: 2px;" alt="" />
</td>For SMF2.0.x (Curve)
<img id="smflogo" src="' . $settings['images_url'] . '/smflogo.png" alt="Simple Machines Forum" title="Simple Machines Forum" />Upload the modified index.template.php over the existing one with your FTP program and your logo image in the forum header should no longer appear.
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Note: The code you must remove may vary from theme to theme. If you cannot find the same code as highlighted here, search for similar code and try to work with that or consult with the theme author for help with the particular theme you are trying to do this for.

How do I remove the Simple Machines Logo?
.
You can also just move that gif file out of the path of the php file.
something like
mv smflogo.gif ./smflogo.gif.bak