What happens to the pressure inside a tied-off balloon as the pressure outside the balloon decreases? I'm almost 100% positive it would decrease, but my meteorology professor (who also teaches physics) says it will stay the same, because no air is escaping the balloon!
Does the rubber of the balloon interfere that much with the equalization of pressure? Is there something else I'm missing? Who is correct?
 A: Pressure inside the balloon as it is brought up to desired interior pressure, let's say at sealevel , is restrained by the exterior press (@14psi). If the exterior pressure drops,  the vessel (in this case a flexible balloon )will expand,  the volume inside increases and the interior pressure will fall. 
So you are  correct,  and should always question any answers by anybody if you aren't satisfied with the answer, or especially the explanation for the answer. 
A: It is well known that for stratospheric balloons, only a small filling at ground level is used because upon ascent the atmospheric pressure decreases and the gas filling will occupy a volume much larger than before. Therefore the pressure in the balloon is approximately equal to the outside pressure and lower pressure gives a larger volume of the balloon.
A: If your balloon is made of very strong material than its volume would not change and the pressure would be the same as your prof states.  But that is not practical.  Also its would be mostly assumed that balloon material is thin and elastic, the pressure inside the balloon is always greater than outside because the elastic exherts a contraction force on the gas inside.  Or an atmospheric balloon is also carrying weight therefore pressure is higher inside.  Helium is very light but can exert a lot of pressure for its mass due to kinetic energy (temperature) hence He balloons float.  
A: If your professor agrees that the gas inside the balloon follows the ideal gas law, then the following applies:
$PV=nRT$     (1)
$nRT$ is constant for the tied-off balloon if the temperature is constant, which it should be.  This means that given the right hand side of equation 1 is constant, the left hand side of equation 1 must also be constant.  Thus, as volume goes up, pressure must decrease to preserve this condition.  Conclusion - if the balloon expands, the pressure must fall as a consequence.
