How do I make a part be shot in the same direction that the player's torso is facing?

While I do not like the method you went about to write this, there were only a couple problems in your code:

• Anchoring the part
• Not setting the MaxExtents
• Not breaking the joints (this fixes some issues with BodyVelocities and parts)

So, in effect, you just need to make sure the part is Unanchored, the max extents are set to Vector3.new(math.huge,math.huge,math.huge), and you need to break the joints of the part.

Here's the edited code:

local player = game.Players.LocalPlayer

mouse.KeyDown:connect(function(key)
    key = key:lower()
    if key == "f" then
        if game.Workspace:FindFirstChild("Pew " .. player.Character.Name) then
            while true do
                wait()
            end
        end
        print("Fired")
        local p = Instance.new("Part", workspace)
        local b = Instance.new("BodyVelocity", p)
        p.Name = ("Pew " .. player.Character.Name)
        p.Shape = ("Ball")
        p.Size = Vector3.new(2.5,2.5,2.5)
        p.Anchored = false -- EDITED
        p.Position = player.Character.HumanoidRootPart.Position + player.Character.HumanoidRootPart.CFrame.lookVector * 5   
        p.CanCollide = false
        b.MaxForce = Vector3.new(1/0,1/0,1/0) -- EDITED // note: 1/0 == math.huge
        b.Velocity = p.CFrame.lookVector * 5
        p:BreakJoints() -- ADDED
        p.Touched:connect(function(part)
            p:Destroy()
            local parent = part.Parent
            if game.Players:GetPlayerFromCharacter(parent) then
                parent.Humanoid.Health = parent.Humanoid.Health - 60
                wait()
            end
        end)

        wait(5)
        p:Destroy()
    end
end)

Here's a much better version of the code using cleaner, more up to date mechanics:

local player        =game.Players.LocalPlayer
local userinput     =game:GetService("UserInputService")

local bulletspeed   =5
local lifetime      =5
local damage        =60
local debounce      =true

local function shoot()
    -- debounce
    debounce            =false

    -- calculations
    local rootpart      =player.Character:FindFirstChild("HumanoidRootPart")
    local rootcf        =rootpart and rootpart.CFrame
    local firedir       =rootcf.lookVector*bulletspeed

    -- rendering
    local bullet        =Instance.new("Part",workspace)
    local vel           =Instance.new("BodyVelocity",bullet)
    bullet.Size         =Vector3.new(2.5,2.5,2.5)
    bullet.Name         ="Bullet"
    bullet.Shape        ="Ball"
    bullet.CFrame       =rootcf*CFrame.new(0,0,-3)
    bullet.CanCollide   =false
    vel.MaxForce        =Vector3.new(1/0,1/0,1/0)
    vel.Velocity        =firedir
    bullet:BreakJoints()

    -- touch
    bullet.Touched:Connect(function(hit)
        local hum       =hit.Parent:FindFirstChild("Humanoid") or hit.Parent.Parent:FindFirstChild("Humanoid")
        if hum then
            local hitchar   =hum.Parent
            if hitchar then
                local hitplayer =game.Players:GetPlayerFromCharacter(hitchar)
                if hitplayer and hitplayer~=player then
                    debounce=true
                    bullet:Destroy()
                    hum:TakeDamage(damage)
                end
            end
        end
    end)

    -- lifetime
    if lifetime and type(lifetime)=="number" and lifetime>0 then
        delay(lifetime,function()
            if bullet then
                bullet:Destroy()
                debounce=true
            end
        end)
    end
end

userinput.InputBegan:Connect(function(input,processed)
    if processed then return end
    local keycode   =input.KeyCode
    if keycode==Enum.KeyCode.F then
        shoot()
    end
end)