The problem is one of the paste drying out in the needle. Putting
the needle under a heat lamp will only make things worse.
An easy & inexpensive solution to the 'plugged needle' syndrome with
pate solder syringes is a trip to the hardware store (take one of
each size needle along) Most hardware stores sell hard steel wire
called 'music wire' in 1 ft - 3 ft lengths. The wire is available in
many diameters. Select the largest diameters that fit in the needle.
Cost of 2, 3 ft long pieces will probably be under $3. When you get
back to the shop, cut about a 3" long piece, fold about 1/4" of
one end over at 90 deg for a handle. Insert it in the needle. Use it
both as a cleaner & as a stopper when not using the syringe.
As someone also suggested, if the paste gets stiff, dried out,
mixing a drop or two of mineral oil with the remaining paste will
probably restore it to the correct consistency. It's the evaporation
of the vehicle that causes the paste to get stiff; the solder & flux
are still fine. Replacing the vehicle, mineral oil, usually restores
the paste to a workable product.
Also storing the paste filled syringes in a zip lock bag between
long periods of non usage will help keep the vehicle from