Mass in a building is good for evening out temperature peaks (buffer effect). But often it is advertised as a way to get free heating energy through heat storage. Well, you can only store, what’s been there before (that even rhymes). Before you can store warmth you would need to have produced it first, otherwise it would be a perpetuum mobile. Thus the only energy that is free to use for storage is solar. If solar radiation impinges on the outer surface of the building, it will there be transformed into heat radiation that will get into the structure of walls and roofs behind this surface. But it progresses in there usually very slowly. Chances are, that it only reaches the upper first centimetres of your construction in winter, before the sun stops shining again. And then what? It all depends on the temperature differences between the indoor and outdoor environment. Let’s say, it’s 10°C outside and 19°C inside. Why would the heat flux go uphill? It wouldn’t. The stored warmth will benefit your garden, and when your walls aren’t insulated, you’ll need to turn up your radiator to maintain even your 19°C. Wind and rain can help to accelerate that process.

However there is a way, how solar energy can actually enter your home, even in winter: through the windows. The solar radiation will then be transformed into heat radiation on the surfaces of your inner walls and floors, which actually benefit your indoor climate for a change. But if your house isn’t insulated, it’ll soon be kiss and good-bye. Maybe, once you finally came home from work, you haven’t even noticed the effect. Plus the only thermal mass that will be beneficial in this process substantially is the thermal mass of your inner structures.
The thermal mass of an average timber frame construction gives sufficient storage capacity to harvest solar energy, but it could be favourable to add a little extra, like (insulated!) concrete floors or double gypsum board layers. Note: with a concrete floor you somehow loose winterly comfort, since concrete is a good heat conductor. It is, so to say, also storing your bodily warmth. If you touch it with bare skin, it feels cold, despite the concrete being insulated towards the ground, because your body heat is conducted away easily. Even if the surface temperature of the concrete floor is above 20°C this goes on, because your surface temperature is way above that as long as you are alive.

I am aware that there are studies out there, that claim that thermal mass is significant in reducing energy consumption in the building sector. That can in some cases be true for cooling energy, otherwise needed in a very hot summer.
The studies I know of are based on empirical information. There is nothing wrong with information gathered by observing things. It is just a question whether these finding can be  you can generalised.

Studies, where houses with the same geometry, but with different amounts of thermal mass and insulation have been tested for their energy efficiency, neglect the effect of termal bypass, which significantly reduces insulating values, and thus can explain the slightly poorer performance of the light weight construction used in the test with the given boundary conditions.  
Peer reviewed research, however, gives a crystal-clear answer to the question: will I feel more comfortable using the same amount of energy in a well insulated house, or in a house with lots of thermal mass and no insulation? Calculations and simulations concerned with this question have been undertaken. The results are unanimous: insulation rules! Thermal mass can be a bonus – if it is insulated from the outside.
But even if you do not want to bother yourself with formulas and math, read the following and decide yourself:

Hot coffee - cold coffee
Coffee in an uninsulated can (only storage capacity) is cold in the fraction of one hour! If we use a cook crate (insulation e.g. from straw) or - more modern - a thermos, then the loss of calorific energy is so small, that the coffee remains hot for hours without energy input.

Dress for warmth or for storage?
Who would in cold weather strike on the idea to put on a knight’s armour (high storage capability) in order to remain warm? For centuries well insulating clothes made of wool, fur or other fabrics with enclosed air layers worked satisfactorily.  They prevent effectively a cooling down of the body.

Sleeping in the bath tub?
Water has a very large heat retention capacity. However if new energy is not constantly supplied to a warm bath (which comes quite expensively) that bath gets rather unpleasantly cold after two, three hours. In the cold season we therefore prefer to sleep underneath thick, well insulating blankets, traditionally. In combination with such a good insulation a -in comparison to the full bath - small hot-water bottle does good services. The energy for this storage must however be paid in this case also (warm water). All-side insulated sleeping bags, which work satisfactorily even at most extreme outside temperatures, are particularly effective – as each outdoor enthusiast or mountaineer would confirm.

Are solid buildings "warmer"?
Such a statement contradicts the experience of all humans, who really ever spent one winter and one spring in a very massive building. Old castles are best examples: with a wall thickness of 80 cm and more these buildings have a high heat retention capacity. This is apparent e.g. in summer, when it matters to keep a balance of day/night temperature differences, as a cooling effect. In winter however it is cold, day and night – hence there is a cold average temperature of these massive storing walls. Old castles could hardly be heated reasonably - the energy consumption is extremely high, but regardless of all the effort the poorly insulated walls remain cold and dry, despite their thickness. Dampness of walls, mould and an unhealthy indoor climate are not rare. A night-time reduction of heating temperatures is completely ineffective. However, a from the outside well insulated "castle" could let the storage characteristics of the large mass become effective.

Excerpt from:
Ist Wärmespeichern wichtiger als Wärmedämmen?
Wolfgang Feist; Passivhaus Institut, Darmstadt; Fachinformation PHI-2000/4 (Kurzfassung)
Translation by Kara Rosemeier

Photo: Kara Rosemeier, Sissinghurst Castle

On Sissinghurst Castle Harold Nicolson, husband of Vita Sackville-West, wrote after a cold spell just before Christmas 1945,  that this convinced him “how bitter it was to inhabit a house which was so cold and draughty and which entailed having to shave in a distant cottage”.