Останавливать ли таймер при выходе из функции или сборщик мусора о нём позаботится?

Это обычный класс и судьба его за пределами области видимости переменной совершенно обычная.

Никаких осложнений типа "вдруг не остановив таймер собьются другие часы", "отвалится коннект к базе данных" или "произойдёт атомный взрыв" не будет - можете посмотреть декомпилированный исходник и глянуть код, этот класс даже не реализует IDisposable, так что сложностей с пониманием никаких нет.

// Decompiled with JetBrains decompiler
// Type: System.Diagnostics.Stopwatch
// Assembly: System, Version=4.0.0.0, Culture=neutral, PublicKeyToken=b77a5c561934e089
// MVID: 988FA076-10C9-4365-AE6D-295A6AA379FE
// Assembly location: C:\Windows\Microsoft.NET\Framework\v4.0.30319\System.dll
using Microsoft.Win32;
namespace System.Diagnostics
{
  /// <summary>Provides a set of methods and properties that you can use to accurately measure elapsed time.To browse the .NET Framework source code for this type, see the Reference Source.</summary>
  [__DynamicallyInvokable]
  public class Stopwatch
  {
    private const long TicksPerMillisecond = 10000;
    private const long TicksPerSecond = 10000000;
    private long elapsed;
    private long startTimeStamp;
    private bool isRunning;
    /// <summary>Gets the frequency of the timer as the number of ticks per second. This field is read-only.</summary>
    [__DynamicallyInvokable]
    public static readonly long Frequency;
    /// <summary>Indicates whether the timer is based on a high-resolution performance counter. This field is read-only.</summary>
    [__DynamicallyInvokable]
    public static readonly bool IsHighResolution;
    private static readonly double tickFrequency;
    static Stopwatch()
    {
      if (!SafeNativeMethods.QueryPerformanceFrequency(out Stopwatch.Frequency))
      {
        Stopwatch.IsHighResolution = false;
        Stopwatch.Frequency = 10000000L;
        Stopwatch.tickFrequency = 1.0;
      }
      else
      {
        Stopwatch.IsHighResolution = true;
        Stopwatch.tickFrequency = 10000000.0;
        Stopwatch.tickFrequency /= (double) Stopwatch.Frequency;
      }
    }
    /// <summary>Initializes a new instance of the <see cref="T:System.Diagnostics.Stopwatch" /> class.</summary>
    [__DynamicallyInvokable]
    public Stopwatch()
    {
      this.Reset();
    }
    /// <summary>Starts, or resumes, measuring elapsed time for an interval.</summary>
    [__DynamicallyInvokable]
    public void Start()
    {
      if (this.isRunning)
        return;
      this.startTimeStamp = Stopwatch.GetTimestamp();
      this.isRunning = true;
    }
    /// <summary>Initializes a new <see cref="T:System.Diagnostics.Stopwatch" /> instance, sets the elapsed time property to zero, and starts measuring elapsed time.</summary>
    /// <returns>A <see cref="T:System.Diagnostics.Stopwatch" /> that has just begun measuring elapsed time.</returns>
    [__DynamicallyInvokable]
    public static Stopwatch StartNew()
    {
      Stopwatch stopwatch = new Stopwatch();
      stopwatch.Start();
      return stopwatch;
    }
    /// <summary>Stops measuring elapsed time for an interval.</summary>
    [__DynamicallyInvokable]
    public void Stop()
    {
      if (!this.isRunning)
        return;
      this.elapsed += Stopwatch.GetTimestamp() - this.startTimeStamp;
      this.isRunning = false;
      if (this.elapsed >= 0L)
        return;
      this.elapsed = 0L;
    }
    /// <summary>Stops time interval measurement and resets the elapsed time to zero.</summary>
    [__DynamicallyInvokable]
    public void Reset()
    {
      this.elapsed = 0L;
      this.isRunning = false;
      this.startTimeStamp = 0L;
    }
    /// <summary>Stops time interval measurement, resets the elapsed time to zero, and starts measuring elapsed time.</summary>
    [__DynamicallyInvokable]
    public void Restart()
    {
      this.elapsed = 0L;
      this.startTimeStamp = Stopwatch.GetTimestamp();
      this.isRunning = true;
    }
    /// <summary>Gets a value indicating whether the <see cref="T:System.Diagnostics.Stopwatch" /> timer is running.</summary>
    /// <returns>
    /// <see langword="true" /> if the <see cref="T:System.Diagnostics.Stopwatch" /> instance is currently running and measuring elapsed time for an interval; otherwise, <see langword="false" />.</returns>
    [__DynamicallyInvokable]
    public bool IsRunning
    {
      [__DynamicallyInvokable] get
      {
        return this.isRunning;
      }
    }
    /// <summary>Gets the total elapsed time measured by the current instance.</summary>
    /// <returns>A read-only <see cref="T:System.TimeSpan" /> representing the total elapsed time measured by the current instance.</returns>
    [__DynamicallyInvokable]
    public TimeSpan Elapsed
    {
      [__DynamicallyInvokable] get
      {
        return new TimeSpan(this.GetElapsedDateTimeTicks());
      }
    }
    /// <summary>Gets the total elapsed time measured by the current instance, in milliseconds.</summary>
    /// <returns>A read-only long integer representing the total number of milliseconds measured by the current instance.</returns>
    [__DynamicallyInvokable]
    public long ElapsedMilliseconds
    {
      [__DynamicallyInvokable] get
      {
        return this.GetElapsedDateTimeTicks() / 10000L;
      }
    }
    /// <summary>Gets the total elapsed time measured by the current instance, in timer ticks.</summary>
    /// <returns>A read-only long integer representing the total number of timer ticks measured by the current instance.</returns>
    [__DynamicallyInvokable]
    public long ElapsedTicks
    {
      [__DynamicallyInvokable] get
      {
        return this.GetRawElapsedTicks();
      }
    }
    /// <summary>Gets the current number of ticks in the timer mechanism.</summary>
    /// <returns>A long integer representing the tick counter value of the underlying timer mechanism.</returns>
    [__DynamicallyInvokable]
    public static long GetTimestamp()
    {
      if (!Stopwatch.IsHighResolution)
        return DateTime.UtcNow.Ticks;
      long num = 0;
      SafeNativeMethods.QueryPerformanceCounter(out num);
      return num;
    }
    private long GetRawElapsedTicks()
    {
      long elapsed = this.elapsed;
      if (this.isRunning)
      {
        long num = Stopwatch.GetTimestamp() - this.startTimeStamp;
        elapsed += num;
      }
      return elapsed;
    }
    private long GetElapsedDateTimeTicks()
    {
      long rawElapsedTicks = this.GetRawElapsedTicks();
      if (Stopwatch.IsHighResolution)
        return (long) ((double) rawElapsedTicks * Stopwatch.tickFrequency);
      return rawElapsedTicks;
    }
  }
}

PS "Класс Stopwatch появился в .NET 2.0 и с тех по не претерпел ни одного изменения."