VHDL Error (10028): Can't resolve multiple constant drivers for net...

let me go through the code, bit by bit. You have other problems other than what you're asking about:

    port
    (
        pin_clk:         in     std_logic;        -- our input clock
        pin_button1:     in  std_logic;         -- user button, fires a signal1
        pin_led_output: buffer std_logic;    -- user sees that a button has been pushed
        pin_ir_output:     buffer std_logic    -- IR output signal
    );
end keyfob_top;

Buffers are not the prefered option of doing things in VHDL now, most people prefer to use an internal signal and assign that to an output. But, I leave it up to you.

-- apply the correct value to button1 (which is a signal)
    procedure set_button1(constant status : std_logic) is
  variable tmp : std_logic;
    begin
  if status = button_up then
    tmp := button_up;     -- has just been pressed
  else
    tmp := button_down; -- has just been released
  end if;
  
  button1 <= tmp;
    end procedure set_button1;

Two things.

1. why dont you do this instead? On real hardware, "status" can only be '0' or '1'.

button1 <= status;

2. Its generally not a good idea to assign a signal thats not in the scope of the procedure. You can do it, but I suspect you'll be getting yourself into trouble, like you already have. Usually, its better to create a signal output on the procedure and connect the signal to that.

procedure set_button1(constant status : std_logic; signal output : out std_logic) is
    begin   
  output <= status;
    end procedure set_button1;
.... then call it:
set_button1(status => input, output => button1);

    -- triggered when pin_button1 has been pressed
    trigger_button1: process
    begin
        wait until pin_button1 = '1';    -- wait until pin_button1 is pressed
        set_button1(button_down);        -- pin_button1 has just been pressed.
        button1_bits_left <= 8;
    end process trigger_button1;

Again, using waits is not the prefered method of clocking logic. You are also using logic as a clock, which is highly inadvisable.

-- do we have anything to send out
    data_out: process
    begin
        wait until baud_tick = '1';          -- line 65    -- wait on rising edge
        -- does button1 need to be serviced
        if button1 = button_down then                
            ....
                    set_button1(button_up);             -- reset our button
                    pin_led_output <= led_off;            -- show user what is going on
......
    end process data_out;
    

There is your problem. You already assigned button1 in the "trigger_button1" process. Signals can only be assigned from a single process, otherwise you get multple driver error like you did. Its because "trigger_button1" is always driving "button_down" onto the signal, while "data_out" always drives "button_up". So they are in conflict.