I live in Ireland and have a horizontal sundial with its gnomon
correctly set up i.e. it is accurately aligned along a true north
south line and it makes an angle to the horizontal equal to my
latitude. My sundial agrees precisely with clock time when I make an
allowance for the equation of time and my longitude.

On the assumption that the sundial remains undisturbed in my garden in
the future, what will happen to its accuracy in one thousand, ten
thousand, one million, ten million years? Presumably tectonic plate
movements will have an effect. What about celestial movements?

I believe the biggest factor will be the precession of the equinoxes,
which will cause long-term inaccuracy, with a cycle of 26,000 years.
http://www.crystalinks.com/precession.html

This is the same thing that causes the Tropic of Capricorn to move 14
meters north per year.

I could not find one mention of how this would affect sundials online,
so I could be wrong...

A sundial is concerned only with the rotation of the Earth on its axis
with respect to a fixed object in the sky, the Sun, and give apparent
time. Clocks keep mean time, a fictitious time measure based on a mean
Sun. The difference in times requires correction, as you note.

The Earth's rotational speed has been slowing since the formation of
the planet. The length of a day, therefore, has been increasing over
geological time. Recently, the length of a day has been increasing by
roughly 2 milliseconds per day. A leap second is accordingly added to
the year approximately every 500 days.

If you want calculations based on that, they can be provided, but the
speed of the Earth's rotation can vary with atmospheric conditions,
and other factors make it impossible to predict exactly how much
variance will occur.

hlabadie-ga

hlabadie--

If a leap second is added to the year every 500 days, that means
length of day is increasing by about 5 microseconds per day, not 2
milliseconds.

United States Naval Observatory

Leap Seconds
http://tycho.usno.navy.mil/leapsec.html

"This scenario is analogous to that encountered with the leap second.
The difference is that instead of setting the clock that is running
slow, we choose to set the clock that is keeping a uniform, precise
time. The reason for this is that we can change the time on an atomic
clock, while it is not possible to alter the Earth's rotational speed
to match the atomic clocks! Currently the Earth runs slow at roughly 2
milliseconds per day. After 500 days, the difference between the Earth
rotation time and the atomic time would be 1 second. Instead of
allowing this to happen, a leap second is inserted to bring the two
times closer together."

hlabadie-ga

hlabadie--

This quote is correct, but you have misinterpreted it.  Yes, the earth
'runs slow' to the tune of 2 milliseconds per day.  That is, rather
than 86400 seconds in a mean solar day, there are about 86400.002. 
This is very different from your statement that "length of day has
been increasing by two milliseconds per day".  That would mean that
tomorrow, the lenght of the day would be 86400.004 sec, and the day
after that, 86400.006 sec.  In reality it has taken 180 years for the
lenght of a day to increase from 86400 to 86400.002 seconds.  The leap
second which is added every 500 days or so is to make up for the fact
that each day is .002 sec too long, it's not there because the days
are getting longer by .002 sec each day.

There is an excellent explanation of this at the very link you posted.
 Here's an excerpt:

Confusion sometimes arises over the misconception that the regular
insertion of leap seconds every few years indicates that the Earth
should stop rotating within a few millennia. The confusion arises
because some mistake leap seconds for a measure of the rate at which
the Earth is slowing. The 1 second increments are, however,
indications of the accumulated difference in time between the two
systems.  (Also, it is important to note that the current difference
in the length of day from 86,400 seconds is the accumulation over
nearly two centuries, not just the previous year.)  As an example, the
situation is similar to what would happen if a person owned a watch
that lost 2 seconds per day. If it were set to a perfect clock today,
the watch would be found to be slow by 2 seconds tomorrow. At the end
of a month, the watch will be roughly a minute in error (30 days of 2
second error accumulated each day). The person would then find it
convenient to reset the watch by one minute to have the correct time
again.

The actual rate of deceleration is on average 1.4 milliseconds per day per century.

hlabadie-ga