How can I safely splice wires inside a lamp?
The requirements for luminaires (lamps, light fixtures) are in Section 410 of the NEC.  In particular, for mechanical strength, we should start with 410.38(A):

Tubing for Arms.  Tubing used for arms and stems where provided with cut
threads shall not be less than 1.02mm (0.040 in.) in thickness and, where
provided with rolled (pressed) threads, shall not be less than 0.64mm (0.025
in.) in thickness.  Arms and other parts shall be fastened to prevent turning.

Your black iron pipe should meet this requirement with ease -- most of it will be Schedule 40, which has plenty of wall thickness.  Just make sure that you have all joints securely tightened (wrench tight, not just finger tight)!
You will also need a bushing (or clamp) of some type to protect the cord from strain where it exits the pipe as per 410.44:

Cord Bushings. A bushing or the equivalent shall be provided where
flexible cord enters the base or stem of a portable lamp.  The bushing
shall be of insulating material unless a jacketed type of cord is
used.

Now, to what you asked about, which starts off in sections 410.22 through 410.24 and 410.28 of the Code.

Luminaire (Fixture) Wiring -- General.  Wiring on or within fixtures
shall be neatly arranged and shall not be exposed to physical damage.  Excess
wiring shall be avoided.  Conductors shall be arranged so that they are not
subjected to temperatures above those for which they are rated.
Polarization of Luminaires (Fixtures).  Luminaires (fixtures) shall be
wired so that the screw shells of lampholders are connected to the same
luminaire (fixture) or circuit conductor or terminal.  The grounded conductor,
where connected to a screw-shell lampholder, shall be connected to the screw
shell.
Conductor Insulation.  Luminaires (fixtures) shall be wired with
conductors having insulation suitable for the environmental conditions,
current, voltage, and temperature to which the conductors will be subjected.

This all should be obvious to you: don't cook, snag, stretch, or snake your wire, and use the right size -- as you mentioned three lampholders, we can presume that you're using medium base as that's what household light bulbs use, which leads us to a maximum of 300W, or roughly 2.5A of current -- 18AWG TFFN fixture wire can handle this with ease (its ampacity under the Code is 6A as per table 402.5), and is rated to 90 degrees C.  Also, wire up your lampholders correctly (reverse polarity lampholders are real people-zappers)!

Protection of Conductors and Insulation.
(A) Properly Secured.  Conductors shall be secured in a manner that does
not tend to cut or abrade the insulation.
(B) Protection through Metal.  Conductor insulation shall be protected
from abrasion where it passes through metal.
(C) Luminaire (Fixture) Stems. Splices and taps shall not be located
within luminaire (fixture) arms or stems.
(D) Splices and Taps.  No unnecessary splices or taps shall be made within
or on a luminaire (fixture).
FPN:  For approved means of making connections, see 110.14.
(E) Stranding.  Stranded conductors shall be used for wiring on luminaire
(fixture) chains and on other movable or flexible parts.
(F) Tension.  Conductors shall be arranged so that the weight of the
luminaire (fixture) or movable parts does not put tension on the conductors.

All in all -- this means you'll be splicing in the base of your lamp, and putting some sort of extra insulation in the pipe to prevent wires from rubbing against the pipe.  (If nothing else, a square or octagonal metal junction box would make a reasonable 'hub' for your lamp base.  Just attach the black iron pipe as if it were conduit, and put a blank metal faceplate on it to serve as a bottom piece.)  The extra insulation material will also get you compliance with 410.18(A):

(A) Exposed Conductive Parts.  Exposed metal parts shall be grounded or
insulated from ground and other conducting surfaces or be inaccessible to
unqualified personnel.  Lamp tie wires, mounting screws, clips, and decorative
bands on glass at least 38mm (1 1/2 in.) from lamp terminals shall not be
required to be grounded.

As to the FPN mentioning 110.14 (which is where the answer to your specific question is, the rest of this is for the next guy from Google who finds this post!), the relevant section is 110.14(B):

(B) Splices.  Conductors shall be spliced or joined with splicing devices
identified for the use or by brazing, welding, or soldering with a fusible
metal or alloy.  Soldered splices shall first be spliced or joined so as to
be mechanically and electrically secure without solder and then be
soldered.  All splices and joints and the free ends of conductors shall be
covered with an insulation equivalent to that of the conductors or with
an insulating device identified for the purpose.
(second paragraph of quote omitted as it is utterly irrelevant for this answer)

So, wirenuts or push-in connectors are the best idea here -- they're identified for the purpose (clearly!), provide their own insulating device, and are trivial to obtain.