How do i make a Dropper that expires?

When creating the dropped blocks, add them to Debris to be deleted after a set amount of time.

function fire()
        local p = Instance.new("Part")
        p.Position = script.Parent.Position
        p.Size = Vector3.new(1,1,1)
        p.BrickColor = BrickColor.new(26)
        p.BottomSurface = 0
        p.TopSurface = 0
        p.Name = "TycoonBrick"
        p.Parent = script.Parent
    game:GetService("Debris"):AddItem(p,5) -- will delete after 5 seconds
end 


while true do 
     wait(2)
     fire()

end 

You should try using the wait() function to wait until we want to remove the part (like you did with the cooldown) and then use Destroy() to get rid of the part.

Try this:

function fire()
        local p = Instance.new("Part")
        p.Position = script.Parent.Position
        p.Size = Vector3.new(1,1,1)
        p.BrickColor = BrickColor.new(26)
        p.BottomSurface = 0
        p.TopSurface = 0
        p.Name = "TycoonBrick"
        p.Parent = script.Parent
    wait(5) --replace 5 with the wait time
    p:Destroy() --call Destroy()
end 


while true do 
     wait(2)
     fire()

end 

Alternatively, you can try to use the Debris method like mattscy did.

function fire()
        local p = Instance.new("Part")
        p.Position = script.Parent.Position
        p.Size = Vector3.new(1,1,1)
        p.BrickColor = BrickColor.new(26)
        p.BottomSurface = 0
        p.TopSurface = 0
        p.Name = "TycoonBrick"
        p.Parent = script.Parent
    game:GetService("Debris"):AddItem(p,5) --replace 5 with wait time
end 


while true do 
     wait(2)
     fire()

end 

Also, I suggest you to read the article about Debris