CFrame Rotation & Position?

Yes you will need to both rotate and position it, as both of these are required to change when something swivels on a point other than it's centre. In this answer I will assume that you are using a single block as a door that you want to rotate about it's Y-axis on one of it's edges. However, it can be adapted quite easily for many purposes.

To accomplish what you want we need to know two things, the position of the pivot point and the offset from the centre of the object you are rotating. The offset can be calculated by taking the inverse of the CFrame obtained from the method of CFrame toObjectSpace from the object and pivot location.

local partCF = part.CFrame
local hingeCF = partCF*CFrame.new(-part.Size.X/2, 0, 0)
local offset = partCF:toObjectSpace(hingeCF):inverse()

Now that we have the offset established and our pivot location is known, we can set up a loop that will interpolate between the start and end positions. In this example, I will rotate a door 90 degrees about it's 'hinge':

local partCF = part.CFrame
local hingeCF = partCF*CFrame.new(-part.Size.X/2, 0, 0)
local offset = partCF:toObjectSpace(hingeCF):inverse()

local alpha = 0
local duration = 2
local elapsed = 0

while alpha < 1 do
	elapsed = elapsed + wait()
	alpha = math.min(1, elapsed/duration)
	
	local angle = alpha*90
	part.CFrame = hingeCF*CFrame.Angles(0, math.rad(angle), 0)*offset
end

Take a look at line 14. If we break this into steps, it will be easy to understand how it works. You start off by setting the part's CFrame to that of the hinges so that it is now centred at the hinge point. Then when you go rotate the part (which is what we do next) the part's axis of rotation is the same as the pivot's. After that, we multiply by the offset again to maintain the same offset between the part and the pivot that we had initially.

If you found any of this confusing, leave a comment and I'll try to clarify.