It may possibly affect the milk supply, but not substantially or permanently.
Engorgement is caused by the increase in milk production, which fill up the breasts. It can also be caused by swollen lymph nodes and excess other fluids as a result of labor and delivery.
Expressing or pumping enough milk to relieve the discomfort was recommended to us not only by a number of websites, but by our own professional lactation consultant (a physician).
However, it's important to realize that since not all the discomfort is being caused by excess milk, expressing milk alone may not be sufficient to relieve the discomfort. As such, you should be careful to not express an excessive amount when trying to relieve discomfort, when other methods may be necessary.
Our physicians have all recommended the continued use of my wife's prescribed anti-inflammatory medication (ibuprofen), and cold/ice packs after feedings, in order to treat the swelling and inflammation not caused my excess milk.
While I've been unable to find any studies or facts regarding the effects expressing milk during engorgement, I was able to find a somewhat related study. Unfortunately, I only have access to the abstract, but it's currently sufficient enough for me as other resources are in agreement.
This study tested the influence of the number of feedings and total feeding times on the milk supply. It confirmed that reducing feedings reduces milk supply, and increasing feedings increases milk supply. What is interesting is that the increases to milk supply were not observed for approximately 48 hours after the increased feeding amounts. I believe this is assuming that the increased amount is held constant (and not done sporadically as you may do to relieve discomfort).
To me, if there is an increase to milk supply, the 48 hours it takes to happen may put the increase after engorgement has subsided for some women.
For our family, we will keep pumping open as an option for relieving discomfort without worry about a cycle of overproduction.