I need to make some axe-eye drifts. But I have been able to open a few eyes with improvised tools like a drift pin or a cold chisel with the appropriate sized shank.
Eye walls can be opened at less than forging temperature because your only bending them not forging them. Forging means changing the cross section of the metal.
Recall that metal bends easiest where it is hottest. So what I do is wrap the bit with a wet rag and then apply heat to one side of the eye wall with a torch. You don't need to get it red hot. If you shined a small area in the center of the eye wall and took that well past blue temper you would be hot enough. I don't bother with shining, I just go by instinct. When you drive your improvised drift that hot side of the eye will open while the cold side resists your drift. I should say, any burrs on the top or bottom of the eye should have been filed off before starting this whole operation.
Once that first side is cooled (you can quench it - it won't harden because you never got it to critical heat) you repeat the process for the other side of the eye. Soon it looks good as new.
Be aware that the fully annealed state you leave the eye in may make it more vulnerable to deformation than it was from the factory. Don't ever use this axe as a wedge. But in every day chopping or splitting work the eye will perform just fine.