Can the CG move out of bounds as fuel is used?
With a straight wing aircraft with all the fuel in the wings, the fuel is close enough to the C of G across the span of the tanks that the C of G shift as fuel is consumed is insignificant.  When this is the case, the weight and balance chart won't have any special requirements to show C of G shift, relative to the fore/aft limits, from fuel burn.

If there is an aux tank that is forward or aft, the effects of its fuel level on C of G will be included in the airplane's weight and balance, and there may be limits on loading of cabin or cargo incorporated into the weight and balance data, and special instructions for C of G calculations when it's being used along with curve added to the wight and balance graph.

On swept wing airplanes, sweep and dihedral in wing tanks causes C of G to move forward as fuel is burned (level drops in the tips first, which are aft).  On swept wing airplanes with a center tank, the center tank is almost always burned first and it causes an aft shift of C of G as its fuel is burned, followed by a forward shift as the wings are consumed.

The C of G charts of swept wing airplanes will include a fuel burn curve to show the C of G location variation with fuel burn. If no center tank, the curve will slant or curve to represent the forward shift, and if there is a center tank it will have a kind of sideways V shape as the C of G moves aft and then forward as the pounds of fuel load declines.

Bottom line is, do whatever the weight and balance chart for the airplane says to do.  If it's something like a 172, there isn't going to be any major shift. If it's something like a Taylorcraft with a nose tank, C of G will shift aft as fuel is burned.  To keep things simple, there may be a baggage restriction that is intended to prevent exceeding the aft limit when the tank is empty at the end of a trip.