Let's have a look at the pathway first (this figure is figure 2 from reference 1), which shows the basic pathway of pigment biosynthesis:

The first important step, the oxidation of the amino acid tyrosine to Dopaquinone (DOPA) by the tyrosinase is the same for both pathways (eumelanin and pheomelanin production) and rate limiting.
The important step is the next one: Either the formation of Cyclodopa (for eumelanin) or Cysteinyldopa (for pheomelanin). As long as the concentration of cysteine in the melanocyte is above 13µM (see reference 1) the formation of pheomelanine is favored and biochemically much faster than the cyclisation of DOPA.
If there are high levels of tyrosine and normal to low levels of cysteine, the cysteine will be depleted at some point and only eumelanin is made.
The opposite will happen if either a lot of cysteine or little tyrosine is available. It will also happen, when the available tyrosine cannot be oxidized into DOPA when there is little or reduced Tyrosinase activity.
And this is actually what happens in the case of the MC1R mutation. MC1R is a receptor on the pigment producing cells which regulates the production of pigment upon external signals (as UV radiation to the skin) (see reference 2 for more details). When mutated is is not or much less functional leading to much less (or no) signals to the downstream gene (MITF) which regulate the enzymes for the pigment production.
Low signals lead to low tyrosinase production which then can only produce little DOPA. Since the cysteine stocks of the cells will not be depleted then, only pheomelanin is produced causing people with MC1R mutations to be red haired and having light skin and freckles.
References:
- Chemistry of mixed melanogenesis--pivotal roles of dopaquinone.
- MC1R, Eumelanin and Pheomelanin: their role in determining the
susceptibility to skin cancer