Yes, NM cable can be in conduit. In fact. NEC calls for it to be in conduit, when protection from physical damage is required.
National Electrical Code 2011
ARTICLE 334 Nonmetallic-Sheathed Cable: Types NM, NMC, and NMS
II. Installation
334.15 Exposed Work. In exposed work, except as provided in 300.11(A), cable shall be installed as specified in 334.15(A) through (C).
(B) Protection from Physical Damage. Cable shall be protected from physical damage where necessary by rigid metal conduit, intermediate metal conduit, electrical metallic tubing, Schedule 80 PVC conduit, Type RTRC marked with the suffix -XW, or other approved means. Where passing through a floor, the cable shall be enclosed in rigid metal conduit, intermediate metal conduit, electrical metallic tubing, Schedule 80 PVC conduit, Type RTRC marked with the suffix -XW, or other approved means extending at least 150 mm (6 in.) above the floor. [ROP 7-94] Type NMC cable installed in shallow chases or grooves in masonry, concrete, or adobe shall be protected in accordance with the requirements in 300.4(F) and covered with plaster, adobe, or similar finish.
There's also some notes in Chapter 9, dealing with how to figure for cables when calculating conduit fill.
Chapter 9 Tables
Notes to Tables
(5) For conductors not included in Chapter 9, such as
multiconductor cables and optical fiber cables, the actual
dimensions shall be used.
(9) A multiconductor cable, optical fiber cable or flexible cord of two or more
conductors shall be treated as a single conductor for
calculating percentage conduit fill area. For cables that
have elliptical cross sections, the cross-sectional area
calculation shall be based on using the major diameter
of the ellipse as a circle diameter.
However, it depends on where the conduit is. If the conduit is underground (or any other damp or wet location), then NM cable is not allowed.
National Electrical Code 2011
ARTICLE 300 Wiring Methods
I. General Requirements
300.5 Underground Installations.
(B) Wet Locations. The interior of enclosures or raceways installed underground shall be considered to be a wet location. Insulated conductors and cables installed in these enclosures or raceways in underground installations shall be listed for use in wet locations and shall comply with 310.10(C). Any connections or splices in an underground installation shall be approved for wet locations.
300.9 Raceways in Wet Locations Above Grade. Where raceways are installed in wet locations above grade, the interior of these raceways shall be considered to be a wet location. Insulated conductors and cables installed in raceways in wet locations above grade shall comply with 310.10(C).
ARTICLE 334 Nonmetallic-Sheathed Cable: Types NM, NMC, and NMS
II. Installation
334.12 Uses Not Permitted.
(B) Types NM and NMS. Types NM and NMS cables shall not be used under the following conditions or in the following locations:
(4) In wet or damp locations
I don't know if you ever got this worked out or not, but I'm in the same boat. Bought an Ecobee3 and I have the RA832A relay. It doesn't appear as though it has any sort of C wire capability.
Awhile back before I bought the unit or knew what my boiler was using in any sort of detail I reached out to Ecobee and this is what they said (I was planning to merge the boiler and AC into one thermostat as they are two separate now).
They said:
The boiler would use Rh and W. The AC would use Rc C G and Y
On their support site they have an article talking about what you can do if you don't have a C wire and one of the options was use a transformer and isolation relay.
Best Answer
In the vast majority of jurisdictions, most DIY work is perfectly legal in a house you own/land contract, and live in. The only exceptions are things like Freon handling, gas lines, and several other crafts, due to the particular hazards (often indirect, e.g. Freon).
You're not allowed to work on houses you don't own for obvious reasons. In many crafts, including electrical, you're not allowed (or permit non-licensed people) to work on houses you rent out, because that would be a huge incentive for "slumlord repairs". Given that rental properties tend to already be at the "more distressed" end of the housing spectrum, this is a recipe for dead tenants. However it is presumed you have incentive to do good work on housing you expect to occupy yourself.
Further, certain minor jobs are always allowed (for homeowners not tenants) due to their simplicity, common-ness, and low chance of critical failure. For instance changing receptacles and switches (typically done for cosmetic reasons) even though this work is harder than it seems. I would certainly expect that changing thermostats on a 24VDC system would be on the short list.
That said, the "only work on houses you expect to live in" principle applies in spades. If you are gussying up a house for sale, STOP. Aside from the "no incentive to do safe work" factor, you're also locking them into choices they may not want. Don't lay white carpets (pretty but nobody wants to live in it), don't fit 20-space stuffed service panels, and don't pick a smart 'stat for them. Give them a price concession of a fraction of the upgrade cost and let them choose.