According to the gzip algorithm, which you should read, compression is done by storing pointers of reoccurring strings back to where they first occurred, using a 32k bytes window. So, basically, it will be unlikely that you will be able to cut out anything from the stream without having to reencode the whole remainder of the stream.

Another problem will be the Huffman-encoded dictionary of literals which will also vary when you cut out stuff from the start of the file.