One of the few ways forefoot running may cause Achilles pain is you may be actually toe striking, not forefoot striking, whereby toe striking means you are landing too high up on your toes without letting your heel drop down to the ground, shown below:
The problem with landing high up on the toes without the heel dropping down to the ground is it increases jarring forces, loading and demands on the calf musculature, while placing enormous strain on the Achilles tendon. The best way around this is to make sure you’re landing properly on your forefoot, and on your toes.
A proper forefoot strike landing is actually a MUCH lower strike angle, striking on the balls, not the toes, of the foot, then the heel is last part of the foot to contact the ground, shown below:
All in all, for some reason, toe striking gets mistaken for being forefoot running when both foot strike types are not the same. For a better demonstration on how to land properly on your forefoot, here’s my YouTube video showing just that!
Also, here are well overt 20 reasons that forefoot running is better than heel strike running!
References:
Nunns et al. (2013). Biomechanical Characteristics of Barefoot Footstrike Modalities. in J Biomech, 46: 2603-2610; 2013.