For reference, here's the Firefox logo:
Even stricter scholars don't include "incomplete images" as haram, e.g., Islam Q&A wrote:
It seems that images which do not have complete features, which do not have a nose or eyes, are not included under the heading of haraam images, and the ones who make them are not included in this warning, because they cannot truly be said to be images, and these images do not imply imitating the creation of Allaah.
In fact, IslamWeb was indeed asked about the Firefox logo (among other logos). Relevant here is:
Firstly, if the image of a face is without a body, it is not included under the prohibited images. Ibn Qudaamah may Allaah have mercy upon him wrote, "The same applies to incomplete images, such as the image of a body without a head, or a head without a body, or an animal's head attached to the body of a non-animal; this is not included in the prohibited images because it is not the image of an animal." [Al-Mughni]
Secondly, there is no harm with the images of animals that are defaced in such manner that their features are unclear. This applies to the bird in the Twitter logo. Ibn Nujaym may Allaah have mercy upon him said, "...likewise, if the image is defaced, it is the same as the image without a head." [Al-Bahr Ar-Raa'iq]
These apply to the Firefox logo. As for other logos, the IslamWeb fatwa describes a logo containing a cross.
If a Muslim cannot remove this cross and there is a necessity or need to use the item bearing its image, then we hope that there would be no harm in using it, following the view of the scholars who held that it is disliked (not prohibited). Some scholars held that it is allowed to use the legal concession in the case of need in order to ward off hardship.
In the context of money with images, Islam Q&A write the problem is so widespread and there is a real need to use it. At some point we have to draw the line, in order to have a functional life. Likewise for logos.