How to measure elapsed time using DateTime class?

Don't use DateTime for measuring time intervals.

Use Stopwatch class, which was designed exactly for this purpose

var clock = new Stopwatch();
clock.Start();
do
{
    // do something
}
while (clock.ElapsedMilliseconds < 10000)
clock.Stop();

Note: Of course you can use DateTime for measuring time, but if you need precision less then second then Stopwatch is right tool for the job.
And Stopwatch have much more readable and easy to use methods for measuring time

In your particular case: "wait for at-least ten seconds from program start, before executing task2" you can use asynchronous approach

var task1 = StartTask1();

await Task.Delay(10000); // next line will be executed approximately after 10 s

var task2 = StartTask2();

Your problem is that you measure the milliseconds part of the TimeSpan. You have to use TotalMilliseconds instead of Milliseconds:

do
{
    Thread.Sleep(100);
    interval = (DateTime.Now - startTime).TotalMilliseconds;
} while (interval < 10000); //wait for at-least ten seconds from program start, before executing task2

With TimeSpan you always have to use the Total properties if you want to know the total time that has passed. All properties without Total in front of them only display the current amount of time that has passed.

For example: A method runs for 1 minute and 7 seconds.

Console.WriteLine(ts.Seconds);

Output:

7

Why is that? Because there are only 60 seconds in a minute and ts.Seconds will start increasing again from 0 to 59.

Compare this to

Console.WriteLine(ts.TotalSeconds),

Output:

67

Now we have the total amount of seconds that have passed, in this case 67 seconds.