Are notions of falsifiability, inspired by the doctrines of the philosophy of science methodologically relevant to economic theory? That is not a simple question. Economic theory is traditionally of a deductive nature: traditionally, the economist starts from certain assumptions, defends them as plausible assumptions, and then proceeds to derive theorems from them. Theorems are considered interesting if they are unexpected, novel, not evident, giving fuel to a debate on the economy, at odds with certain established views on the economy and the like. This is called Deductive Model Exploration, more fully explained if you click the link, but here are some first examples:

Example 1: Adam Smith, one of the founders of economics in the 18th century, spends some pages to convince the reader of the plausibility of the assumption that entrepreneurs invest in the most profitable industry the can find. He does the same with the assumption that if more of the produce of that industry is brought to the market, its price will go down, and if less, it will go up. From this he deduces that if some industry yields, as a result of high prices, a more than average profit, this will attract investments withdrawn from other industries. These will increase the industry's production, thus lower the price of its product thereby lowering the industry's profit. The striking conclusion ("theorem") is that the profits of all industries in a country tend equalize.

Example 2: Many simple and instructive examples of the procedure of deduction in theoretical economics, can be found in a classic and still higly renowned book by Paul A. Samuelson called Foundations of Economic Analysis, Cambridge, Harvard University Press, 1947. A lot of his deductions in this beautiful book proceed from the assumption of stability: he assumes (pp.262-3) that normally markets are like the left type of figure 3 rather than the right type. From that he proves mathematically the theorem that an increase in demand (a rightward displacement of the demand curve as a result of, say, a change in consumer taste) will definitely lead to a rise of price. In this book he intends to show by many examples how mathematical deduction can lead from plausible assumptions to a meaningful theorem.  

Many famous results of economics are called theorems: the Heckscher Ohlin theorem, the factor price equalization theorem, the Modigliani Miller theorem. Often the are called effects: the Keynes effect, the real balance effect, but this also refers to theorems.

The assumptions from which theorems are proven are of several different type. Firstly, there are basic assumptions characterizing the "research programme" or "paradigm" involved. A group of economists proceeding from a shared set of basic assumptions is called a "school" or "current". Terms like Neoclassical, Keynesian, and Monetarist refer to such schools. Secondly there are application assumptions needed to employ the basic assumptions to some specific field of application, like consumer behaviour, international economics, portfolio investment, the labour market  etc. In all these fields of application one can be a "Keynesian", "Monetarist", "Neoclassical economist", etc. Finally you have special assumptions, usually of a technical nature, often highly simplifying assumptions needed to keep control over the mathematics of the problem and be able to derive the desired theorem. Take the first version of the Stolper Samuelson theorem stating that if, for instance, as in the US, labour is relatively scarce, the workers will absolutely benefit from any tariff on imports. Stolper and Samuelson had to assume that the country imposing the tariff is infinitely small, and thus has no influence on world market prices. That is an example of a special assumption. Not much later, Metzler stated the conditions under which the same is the case when there are two countries, neither of them infinitely small. That is an example of relaxation of a special assumption.

Economists working on deductive model exploration "falsification" is too simple as a criterion: they know their models are abstractions and a host of aspects of them are simply false empirically. They know, but yet they proceed, because they are not after a true model, but after instructive, relatively simple models that entail interesting consequences for the economy, and the want to find out what these consequences depend upon.