Do all chemically-fueled rockets need foam insulation? Has it always been this way? Is damage to the rocket or space plane still a worry?
The Space Shuttle used liquid hydrogen, contained in the external tank along with liquid oxygen. While the Falcon rockets do use liquid oxygen, they do not use liquid hydrogen. Keeping the liquid hydrogen cool was the primary driver for the foam.
The Orbiter was mounted alongside and below the the top of the external tank. The Falcon payload is mounted at the very top of the launch vehicle. Even if the Falcon rocket did use foam, and even if a piece of foam broke off, it would not strike the payload as did the piece of foam that broke off of the Space Shuttle external tank.
Do all chemically-fueled rockets need foam insulation?
Not all, but some do. Generally the small subset of ones that utilize hydrogen fuel. The hydrogen-fueled Delta IV uses essentially the same insulating foam as shuttle did.
Photo from clickorlando.com
The hydrogen-fueled SLS uses a similar foam.
Photo credit NASA
This question Insulation on rockets--why take it to space? and its answers discuss insulation on rockets, including some that jettisoned it on purpose.
As the other answers say, impacts from shed insulation is typically not a problem since current boosters don't have glass-covered spacecraft mounted to their sides.
Many un-insulated rockets shed tremendous amounts of ice at liftoff. Ice is much denser and more dangerous than insulation. This is also generally not an issue.
A still from this video shows vast quantities of ice falling off a Saturn V at liftoff.
I don't have a source sorry, but I suspect the reason for having insulation is to prevent formation of liquid oxygen on the out side of hydrogen tanks.
Hydrogen is so cold that it can cause the Oxygen (and Nitrogen) in the the air to condense. Liquid Oxygen just dribbling onto the launch pad and rocket would be a bad thing. Steel burns in liquid oxygen. Decidedly unhealthy.
Insulation would help prevent boil off during flight. However I suspect if you did the math, the weight of the insulation is probably put to better use by just adding more propellant into the tanks.
Insulation would also help prevent boil off when on the pad. But the cost of a little more LOX added to top off the rocket (or chilling with bubbled cold helium), would be less than the cost of insulation and lost lift capacity lugging insulation up into space.
Any ice build up, just shakes off once the rock launches, and is a feature in a way. Ice is a moderately good insulator, free to install and removes itself at launch automatically.
So in conclusion you need insulation on liquid hydrogen tanks, because they are much colder (the outside is relatively hotter) and it prevents dangerous LOX build up. However for RPX/LOX rockets insulation is dead weight, and topping off (or chilling with bubbled cold helium) the tanks prior to launch is more economical