Whenever I am trying to increase the resolution of the logo I am exporting from Illustrator the file size gets increased automatically, I would like to know if I could increase the resolution while the file size remains the same.
Increasing the resolution is literally creating your image with more pixels. More pixels = more data = bigger file size. There is no way around that. You can of course compress raster images, but without compression there is no way to get around an increase in file size (if there was, compression wouldn't even be a thing).
Since you're working in Illustrator you may be better off using a vector format (in which case resolution isn't an issue). For more on that see:
For more on compressing your raster image, see some of the many previous Q&As on image compression and file size: