why flat priors and noiseless data are required for MAP learning The brute force Maximum A Posteriori estimation involves computation of posterior probability for all values of $\theta$ and then we choose the value of $\theta$ that maximizes $P(\theta | D)$. Right.
In this concept, why is it necessary to have uniform priors and noiseless data (as mentioned in one of the articles that I went through)?
I do not see any point in having such assumptions in place. For example, even if priors are non-uniform the posterior probability given by $p(\theta|D)$ can easily be obtained. Further, even if the data is noisy, we can always model this noise by some distribution (depending upon the source it comes from) and the parameters of this noise distribution could be appended to form extended $\theta$. Then, we can easily compute the posterior probability by Bayes rule.
Your take on this ?
 A: 
In this concept, why is it necessary to have uniform priors and
  noiseless data (as mentioned in one of the articles that I went
  through)?

No and no.
Maximum a posteriori estimation is estimation procedure aiming at finding the maximum (mode) of the poisterior distribution
$$
\begin{align}
\hat{\theta}_{\mathrm{MAP}}(x)
&= \underset{\theta}{\operatorname{arg\,max}} \ f(\theta \mid x) \\
&= \underset{\theta}{\operatorname{arg\,max}} \ \frac{f(x \mid \theta) \, g(\theta)}
  {\displaystyle\int_{\vartheta} f(x \mid \vartheta) \, g(\vartheta) \, d\vartheta} \\
&= \underset{\theta}{\operatorname{arg\,max}} \ f(x \mid \theta) \, g(\theta)
\end{align}
$$
to find it, you need likelihood function $f(x \mid \theta)$ and a prior $g(\theta)$, any prior. There is no reason whatsoever why the prior should be uniform. With uniform prior $g(\theta) \propto 1$, MAP reduces to maximum likelihood estimation since you are finding the maximum of $f(x \mid \theta) \; g(\theta) = f(x \mid \theta) \times 1$. With non-uniform prior, you simply include such prior information into your model. If prior needed to be uniform, then you wouldn't be able to use non-uniform priors, so the method would be pretty useless for Basyesian estimation.
You answered your second question yourself

Further, even if the data is noisy, we can always model this noise by
  some distribution (...)

In real world there is no such a ting as a "noiseless" data, at least I never seen it besides the syntetic, simulated data.
