Set attributes like this
start.setAttribute('readonly',true)
function func()
{
var selectboxlist=document.querySelector('select');
if (selectboxlist.selectedIndex == 2 || selectboxlist.selectedIndex == 13 || selectboxlist.selectedIndex == 14) {
var start=document.querySelector('.start')
start.value = "00:00"; // works
start.setAttribute('readonly',true) // doesnt work
console.log(start)
}
}
<input type="time" class="start" />
<select class="selectboxlist" onchange="func()">
<option>1</option>
<option>2</option>
<option>3</option>
<option>4</option>
<option>5</option>
<option>6</option>
<option>7</option>
<option>8</option>
<option>9</option>
<option>10</option>
<option>11</option>
<option>12</option>
<option>13</option>
</select>