Yes, magnetic fields can affect chemical reactions. These are usually reactions in which two radicals are produced. The reaction of a molecule R-R can form two radicals and is written generically as R-R $\rightarrow$ R. + R.. The two radicals each have an unpaired electron. As the radicals are produced in close proximity, because a chemical bond is broken to produce them, the pair is called geminate. These two radicals can then recombine and so form the chemical bond again. In this case the spins are paired, conventionally written as $\uparrow \downarrow$. However, if the spin of one is changed then instead of recombining as a singlet a triplet is formed, again conventionally $\uparrow \uparrow$ and the triplet state may undergo different reactions and not reform the starting compound.
The triplet state, as the name suggest has three substates which are of equal energy (degenerate) in the absence of a magnetic field. However, two of these states are sensitive to a magnetic field, one rises in energy, the other falls by an equal amount so that the total energy is constant, (see sketch). The triplet has a higher energy than the singlet, which, as it is spin paired, is insensitive to magnetic fields.
Normally the difference in energy is quite large compared to thermal energy and this makes the S-T crossing very slow and is unimportant in the absence of a magnetic field. (The energy gap at zero magnetic field is 2*J* where J is the exchange energy); see sketch. As can be seen there is a magnetic field at which S and T cross (dotted circle) and this will facilitate rapid crossing to triplets and thus to the possibility of different chemistry. As the crossing is quite narrow, in terms of the range of magnetic field strength, only if the magnetic field is of just the right value will S-T crossing occur.
The field strength needed to cause S-T crossing is typically of the order of a few hundred Gauss ($10^4$ gauss = 1 Tesla), far stronger than the earth's magnetic field, (approx $5.510^{-5}$ Tesla) thus there is still debate as to whether smaller fields can cause a change in chemistry in people. This may be possible as many reactions occur in enzymes and these are far more complex than simple solution phase chemistry.
(For the sake of completeness note that there are other types of reactions that are sensitive to magnetic fields such as electron transfer and triplet - triplet annihilation )