If we take as first tentative definition of the action on the string world-sheet the Nambu-Goto action then the metric on the world-sheet is not dynamical and it is simply the induced metric from the Minkowski space-time. In particular, if the strings really propagate in a time-like direction, this induced metric is Lorentzian, i.e. of signature (-+).
As we do not know how to make sense of the Nambu-Goto at the quantum mechanical level (because of the square root), we try to replace it by the Polyakov action. In this formulation, the world-sheet metric is a dynamical variable and is equal to the induced metric only when the equations of motion are satisfied. In any case, this metric should still be Lorentzian.
(Remark: the replacement of the Nambu-Goto action by the Polyakov action is generally justified by saying that the two actions define the same classical theory. But if the Nambu-Goto theory was defined at the quantum level, we would like to have an equivalence of quantum theories. Even if we don't know how to define the Nambu-Goto theory at the quantum level, it is possible to argue that if it was defined, the quantum equivalence should hold: see for example Chapter 9 of Polyakov book "Gauge fields and strings").
Now, to define a quantum theory from the Polyakov action, we should have to sum over the world-sheet topologies and to integrate over the space of Lorentzian metrics on a given world-sheet topology. In other words, we are trying to define some form of 2-dimensional quantum gravity. Using the Weyl invariance of the Polyakov action and assuming that the Weyl anomaly vanishes (critical dimension), we can reduce the integration to the space of conformal classes of Lorentzian metrics. The problem is that we don't know how to define such integral. One first problem is that the space of conformal classes, as the total space of metrics, is infinite dimensional in general (this is the case for a case as simple as a 2-torus). A second problem is that a Lorentzian metric on a world-sheet would be in general singular at the points where the strings break and join (the only compact orientable surface which admits everywhere defined Lorentzian metric is the torus). A still subtler fact is that Lorentzian geometry depends heavily of the degree of differentiability : for example two Lorentzian metric can be conformal equivalent by a 1 times differentiable map but not by a 2 times differentiable map, so it is not even clear what is the correct mathematical meaning of "conformal class" for the physical theory. These three problems : infinite dimensionality, singularities, dependance on regularity are typical of hyperbolic questions.
As we don't know how to define the integral over the space of Lorentzian metrics, we can try to do a Wick rotation and to integrate over the space of Euclidean metrics, i.e. of signature (++). Even if we don't know how to integrate over the space of Lorentzian metrics, we would like that if it was defined, it would give the same result than the integration over the space of Euclidean metrics. An heuristic argument for that is given in 3.2. of Polchinski book. An easy but still convincing point is that the usual problem with the Wick rotation for quantum gravity in higher dimension (the wrong sign of the cinetic term of the conformal mode) does not appear in the two dimensional case considered.
So we are now trying to define the integration over the space of conformal classes of Euclidean metrics. The point is that the three problems of the Lorentzian case disappear: in general, the space of conformal classes of metrics will be finite dimensional, there exists many metrics without singularity and there is no regularity problem. These three facts: finite dimensionality, smoothness, regularity, are typical of elliptic questions.
To describe more precisely the space of conformal classes, we have to say what are the topologies considered. The goal of a perturbative theory is to define the scattering amplitudes, S-matrices, between states in the spectrum of the theory coming from and going to infinity. In particular, in the closed string case, the most obvious picture of world-sheet we want to consider is a oriented two dimensional surface with some holes in the middle and with some number of cylinders going to infinity, the external states being inserted in these cylinders. Now, as the theory living on the world-sheet is conformal, the state-operator correspondences implies that to insert an external state in a semi-infinite long cylinder is equivalent to suppress the cylinder, replace it by a marked point and insert an operator, called the vertex operator associated to the external state, at this marked point. After this reformulation, it is only necessary to consider compact surfaces with marked points and the space of conformal classes of metrics on such topology is finite dimensional. Once the measure to consider on this finite dimensional space is determined, perturbative string theory is well-defined. There is a similar story for the open string case.
What is usually called a Riemann surface in mathematics is a surface with a complex structure. By a well-known theorem, the data of a complex structure on a oriented surface is the same as the data of a conformal class of metrics. In other words, the space of conformal classes of metrics is really the same as the space of Riemann surfaces. (In physics, it happens that some people use the expression "Riemann surface" for just surface (=smooth two dimensional manifold) and I think it can be confusing).
I don't really understand the last part of the question because I don't know what is the meaning of "Riemann surfaces". If it is just smooth surface, the question (in the closed case) would be : why do we have to include surfaces with an arbitrarly number of holes? The answer is that from the moment you allow string to interact by joining and splitting, you can construct any surfaces by a sequence of these interactions. If the meaning of "Riemann surfaces" is surface with complex structure, the question would be : why do we have to integrate over the space of complex structures? The answer is that perturbative string theory is some version of 2d quantum gravity coupled to matter and so we have to integrate over all metrics, integration which reduces by conformal invariance to the space of complex structures.
Conclusion: the short answer is the same as in the ahalanay's answer: No, simply because one side of the relation, quantum Lorentzian world-sheet, is not well-defined. Still, there is a general logic (that I have try to explained above) that to work with a quantum Euclidean world-sheet gives the same results that a quantum Lorentzian world-sheet would give.
Addendum: if one really looks to the details on how the integration on the moduli space of the conformal structures should be done, one finds that although the world-sheet has generically Eucildean signature, it has Lorentz signature when a string is going on-shell: see http://arxiv.org/abs/1307.5124