I look at it this way: they’re meant to serve as a reminder, and they’re a good one, so it is OK to leave them on. Most of the fasting you do during Lent won’t be marked with ashes on your forehead.
On the other hand, the readings for Ash Wednesday include this passage: “When you fast, do not look gloomy like the hypocrites. They neglect their appearance, so that they may appear to others to be fasting. Amen, I say to you, they have received their reward. But when you fast, anoint your head and wash your face, so that you may not appear to be fasting, except to your Father who is hidden. And your Father who sees what is hidden will repay you.”
I take that to mean that if you are concerned you could seem to be showing off that you’re observing Ash Wednesday, it is also OK to wash the ashes off. Many of us don’t even get them until the evening Mass, after which we go home, wash our faces, and go to bed. You can go either way; either way is OK.