It is not ok to stare at the fire because the high temperature will evaporate your water in the eyes. If you stare a long time, the eyes may feel dry and uncomfortable. What's worse, your eyes may get into the ashes from the fire. If you wear the contact lenses, you should better not stare at the fire.
Well, generally speaking, staring at fire will not damage your eyes' vision, but it can damage your eyes in some situations. For example, if you look at the fire in a far place, definitely, it will be ok. But when you stare at the fire in front of it, it can be dangerous for your eyes. You should know that despite being inside of your eye, there are cells that normally never see light. So when you go from a low light area to a bright area, your pupils are still dilated. And then the light entering is not focused onto the cells in the back of your eye that are designed for seeing, and light falls on these other sensitive cells causing the pain that you feel. In a long effect, it will hurt your eyes. So just be careful about it.
The human's eyes have some capability of adapting to the rays including the sunshine and fire. However, if the rays are strong for the eyes to stand, the eyes will feel photophobia and uncomfortable. The normal reflection is to narrow the eyes or use other article to segregate the severe rays to relief the unwell feeling to the rays. The sudden photophobia happen, is really an alert to you that your eyes should be protecting now. Maybe the rays are too strong to stare at! Usually, natural fire will be not such strong enough to hurt your eyes immediately if you not staring at it for a very long time. If you have to do like this in some conditions, a pair of sunglasses will be recommended to you to protect your eyes.