It is consistent without the axiom of choice that $\mathcal P(\mathbb R)$ cannot be linearly ordered. Since $\mathbb R^\mathbb R$ has the same cardinality as $\mathcal P(\mathbb R)$ this means that it is consistent that the set of all functions from $\mathbb R$ to itself cannot be linearly ordered.
Assuming the axiom of choice, of course, this set can be linearly ordered in a myriad of ways (well order; dense order; etc.)
Addressing the issue in the third comment, "Is it neccessarily false if AC is false", the answer is no in a very trivial way. This is quite a local requirement. We can ensure that the axiom of choice holds for sets like this, and even much much larger, but fails miserably in many many ways much later in the hierarchy of sets.
Also, it is known that the assertion "Every set can be linearly ordered" does not imply the axiom of choice. That is to say there are models in which choice fails, but every set can still be linearly ordered. In particular, this one.
Also related: Total order in the power set of the real line
To give some more details:
It is impossible to prove without some choice that $\mathbb R$ can be well-ordered, therefore it is impossible to prove that $\mathcal P(\mathbb R)$ can be well-ordered as well, unless we assume some choice holds.
It is consistent that the axiom of choice fails, but every set can be linearly ordered. This is true in Cohen's first model which is a model in which we begin from the constructible universe, add a countable set of Cohen reals and use finitely supported permutations of $\omega$ to define an inner model of the forcing extension, in which the collection of the generic reals is a Dedekind-finite set.
In this model every set is definable from an ordinal and a finite set of real numbers, and therefore we can define a linear order of the universe, and in particular we can linearly order any set -- including $\mathcal P(\mathbb R)$. The proof that this property holds is very difficult and far from intuitive.
It is consistent that the axiom of choice fails, and there exists a subset of $\mathcal P(\mathbb R)$ which can be partitioned into countably many pairs, and we cannot choose a point from each pair. Such set cannot be linearly ordered, and therefore $\mathcal P(\mathbb R)$ cannot be linearly ordered in such model.
One model in which this occurs is Cohen's second model, which is constructed in a similar way as before, only we take a different permutation group and using it we generate not only Dedekind-finite sets of reals, but we generate enough of them to accommodate the elements of the pairs mentioned above.
It should be remarked that Monro proved that one can actually force above Cohen's first model (or rather its "evil twin", the Halpern-Levy model) that a set that cannot be linearly ordered exists in $\mathcal P(\mathbb R)$, however it is a different set with different properties.
If we do not assume anything about the axiom of choice, it might hold and everything could be linearly ordered; or it might fail at a stage far beyond the real numbers. An example of such model is the model we get from another paper by Monro, this time we perform a very similar construction to Cohen's first model only we use a larger cardinal rather than adding subsets to $\omega$, and if we are careful enough we can ensure not to add any new sets to $\mathcal{P(P(P(P(P}(\mathbb R))))$ and therefore preserve the well-orderability of $\mathcal P(\mathbb R)$, while the axiom of countable choice fails.
