Your best bet is setup a new document in Photoshop, to the size you want this graphic to be in pixels, with 72dpi, then copy and paste your logo from Illustrator to Photoshop as a smart object or just pixels. Then from Photoshop click File > Save for Web > and then select gif.
Your problems have probably come from the image size being too large for the space you've given it, and this reduction is causing the display problem
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