### In QGIS
If you want to do this in QGIS, the "Freehand raster georeferencer" might work. You can add it via "Plugins -> Manage and Install Plugins...". It allows you to move rasters around, using the mouse. It did work for me for some GeoTiffs, but failed to open others.

### From command line
The other option is to use GDAL functions from the command line. You need to have GDAL with OSGEO installed on your computer for this to work.
You can move a GeoTiff by setting (overwriting) new upper left and lower right coordinates of your geotiff using `gdal_edit.py -a_ullr`. I recommend you make a backup of your GeoTiff before you start, as the command diretly modifies the file.  
If you do not know the current coordinates of the corners of your geotiff, run `gdalinfo your_map.tiff` and look for the "Upper Left" and "Lower Right" cordinates (first pair of numbers, which are in metres, if you geotiff's LENGTHUNIT is metre). Keep a note of those numbers, and add however much you want to move your raster towards east (subtract 100 to shift 100m westward) to the first number of *both* corners and do the same for shifting north with the second number of both corners. Then call `gdal_edit.py -a_ullr new_ulx new_uly new_lrx new_lry your_map.tiff`.  
Import the GeoTiff into QGIS again and you should see it in its new place.