Reincarnation - the idea that the human soul can be reborn into the physical world in another physical body - is explicitly rejected by the Ismaili Imams and thinkers of the Fatimid period and thereafter. 

It is important to note that even the concept of reincarnation in Hindu tradition seems to be a later development and not something rooted in the original Vedic religion which was its predecessor. (Laumakis, Stephen J., An Introduction to Buddhist Philosophy, 2008)

The Ismaili Ginans composed in South Asia do mention reincarnation, but not as a doctrinal teaching or as a fact. Reincarnation in the Ginans is evoked always as a negative outcome – in the sense of a warning or a threat – if one does not recognise the Imam of the time. Reincarnation in the Ginans is evoked as a symbol of perpetual imprisonement and not asserted as an Ismaili doctrine. On this point, Aziz Esmail - a scholar of South Asian Ismaili literature - remarks on the subject of rebirth in the Ginans deserve reflection:

“The doctrine of reincarnation is a very old one, as is evident in its presence in cultures across the world from Ancient Greece, Africa, through to India. It is not to be found, of course, in scriptural traditions in the Semitic languages…  Like Arabo-Persian Islam, Arabo-Persian Isma‘ilism is reported to have vigorously repudiated this doctrine. The Fatimid Caliph al-Mu’izz is reported to have sharply REPRIMANDED one of his missionaries in Sind for tolerating the belief.”
– Aziz Esmail, (A Scent of Sandlewood, 65)

“By the time we come to the Ginans, however, we find this theme very much part of their texture. Perhaps ‘theme’ is a wrong word. The doctrine is not canvassed or urged – it is a given, taken for granted. It is part of the atmosphere from which the literature inhales its substance.  The Ginans invoke the doctrine; they do not assert it. An asserted idea is the object of thought; an invoked idea is a means, often unconscious, to some other conscious object or purpose.”
– Aziz Esmail, (A Scent of Sandalwood, 65) 

“As it happens, the Ginans are eclectic rather than exclusive in this regard. They mention not only reincarnation, but also what we might call the prophetic model of life after death. Both are present without contradiction, without straining after logical resolution… In the Ginans one is told that to know the truth, which can be only through the true faith, is necessary for smashing, once and for all, the relentless chain of karma, of action and its consequences, which make the wheel of rebirth spin on and on. In this way, the idea of re-birth is invoked not to promote it as an object of belief, but to promote commitment to the true faith…  The other model – what we call the ‘prophetic model’ – with its emphasis on judgment in the hereafter by God, and of paradise and hell as alternative destinations, exists harmoniously, side by side, with the doctrine of reincarnation.  In fact, the prophetic model is more pronounced, more consciously elaborated. Reincarnation occurs in a formulaic form. But the other idea is more often elaborated, and frequently in vivid, graphic terms. We are never presented with an anecdote or a descriptive account of a soul caught in the cycle of rebirth. We are only reminded, in passing, of the misery of such a fate. But the other idea, divine judgment in the hereafter on one’s conduct in the world, forms the subject of the whole of a short Ginan and of many verses in numerous other hymns.”
– Aziz Esmail, (A Scent of Sandalwood, 66-67)

The most recent guidance of the Ismaili Imamat on the subject of reincarnation comes from Imam Sultan Muhammad Shah. Below is the question and answer to this question posed by the Ismaili Missionaries in 1954 and the Imam's response:

On 12th February, 1954, Count Paroo wrote to Khudavind asking following question: “Do the Ismailis believe that there is re-birth on this earth to repay and receive repayment of Karmic debt or do we believe that this re-birth will be in a (higher) creation than the human beings?

The Imam replied: “Obviously reborn means in a higher sphere than this earth. Without going to the final spiritual sphere there will be further triumph before the highest points are reached unless those highest points are reached in this world and on this earth by the general rules of the Ismaili faith beginning with kindness, gentleness, etc and going up to highest love of union with Imam.” 

Source: Subjects Discussed By The Religious Study Group of Mombasa, Count Paroo, 12th February, 1954

The Hindu and Buddhistic explanations of life after death, with always the influence of the soul taking forms either much lower, such as the lowest animals, or much higher, like some so-called Gods of both Brahmanism and Buddhism, seem to many brought up outside their immemorial tradition as more a hope and pious wish than anything founded on fact....This is a third way of looking at survival after death (apart from the Biblical raising of the body, and from the indefinite and varied doctrines of the several Hindu schools of thought and the two great Buddhist Northern and Southern sects). It is the hope of all true Muslims, like their Prophet, namely, "Companionship on High."
- Imam Sultan Muhammad Shah, " Reincarnation or Companionship on High?" in Africa Ismaili Kisumu Supplement, March 28, 1969 (http://www.amaana.org/sultweb/companion.htm)

The Imam's response denied rebirth back into this physical world and stated that the human soul is only reborn into a sphere or plane higher than or beyond the physical world. More on the nature of the Afterlife and the higher planes of being than this physical world is explained here: What Happens to the Soul After Death?