The date picker itself currently doesn't have that capability (please create a new feature request in the PowerApps Ideas page for that)
What you can do is to notify (visuall) the user that this is an invalid entry. For example, in the app shown below, when the date is selected the border of the picker is changed and an error message is displayed:

This can be accomplished by setting the properties below:
DatePicker1.BorderColor: If(DatePicker1.SelectedDate > Today(), Color.Red, RGBA(101, 128, 187, 1))
Label1.Text: "Dates in the future cannot be selected"
Label1.Color: Color.Red
Label1.Visible: DatePicker1.SelectedDate > Today()
Notice that you can also use similar expressions to prevent the user from submitting a form (in the element that causes the form submission, you can set its 'Disabled' property to DatePicker1.SelectedDate <= Today(), for example).