Something sounds a little fishy about the techs you've gotten.
See if you can't get a recommendation from a neighbor?
A name doesn't make a competent furnace repairman: e.g. Agway
is big around here, but don't let them touch your furnace.
Anybody can get a job there.

....
After one visit there was a reccomendation to get a new tank.
The
owner of the company says that sludge is collecting and this is
the
problem.

Sludge collecting does NOT mean you need a new tank!


But last week a technician came it to start the heat (since it
had
locked up) and could not see anything wrong. He was there for
quite
awhile trying to come up with ideas about my problems.
...

And then the heat stopped again the next day.

So my question is: If there was sludge in the tank (1)
wouldn't it
show up in the filter;

Yes, it certainly would show up in the filter. And in the
nozzle. Sludge in a tank is common and unless you let the tank
get real low, usually isn't a problem.
Sometimes sludge will cause a problem right after a fillup; it
gets stirred up and riled up and you'll find it in the filter.
But if the filter does its job it doesn't get to the nozzles or
the pumps.

and (2) wouldn't it take a while to cause
problems again after servicing.

No, not necessarily. But maybe g. Or NOT! In other words, it
depends on too many things to say one way or the other.


Right now the heat goes off about once a day, and we hit the
button
(just once) to get it going again. Takes 5-10 seconds to fire
up.

Someone said that was too long, but in reality it is not. We
have three oil units: 2 furnaces and a water heater; they all
take 5 to 10 seconds to ignite.


Since a new tank is going to be about $2000, I'm obviously
looking for
any other possible solutions.

Understandably. If there IS too much sludge in the tank, they
can be cleaned out; not necessary to replace a perfectly good
tank because of sludge. It sounds, however, like the filters are
staying clean. How many nozzles have they changed? If only one
or two, then forget this path.
Check the nameplates on the burner and look for the nozzle
orfice specs: Angle and gallons/min (gpm). Those are important
and normally MUST use the right nozzle to ignite properly and
reliably. If no one has checked, it could be as simple as the
wrong nozzle; especially if all they do is look at the old nozzle
to see what ot use for the replacement: It has to be the right
nozzle.

Go down and watch the furnace fire up and turn off a couple times
while someone operates the thermostat for you. Observe whatever
happens and report back with that info. There's a little door
you can open to see the fire inside after it lights; the flames
should be bright, constant, and quite yellow in color. No smoke,
no puffing, etc..


Also:
The tank is inside the basement.

I had flooding in the basement in the fall, but I also had
recurring
burner problems last winter, no problems the previous 10-12
years.
I have a tank-less hot water system in the furnace.
I've also noticed that one of the heating zones is not working
(I have
two zones with hot water pipes -- one of these is not working,
one zone
with old radiators). The technician said this must be a short
in the
thermostat.

Jim

No idea what to say about those things; not enough details. YOu
need to gather together as many details as you can for folks to
make eudcated guesses about the problem/s you have. Personally,
I don't think you've had a good tech come out there yet -
quitting that often, it shouldn't be that hard to pin down.

If it's excessively cold in the basement, say below freezing, or
drafty, you might try switching to #1 fuel oil next time you get
down to about half a tank and see if htat helps. I'm assuming
you're burning #2 fuel oil; #1 is lighter and cleaner. It if
helps, try another tank. If it keeps working, after a tank or
two, go back to #2 and see if the problems return.

You could also have air in the oil lines, lots of other things,
but without knowing what the feeds are, oil line returns, if any,
etc., it's real hard to guess much further.

HTH,

Pop