There are six approved litanies within the Roman Rite of the Catholic Church. There are no officials rules as to how to pray any of these litanies. That said, however, there are a few traditional ways of praying them.
For example the Litany of the Saints is usually chanted while kneeling during the Mass of Ordination or a solemn profession of a religious. It may be chanted or recited while making a procession as on Rogation Days.
The Litany of Loreto may be said privately or in public as a novena either recited, sung or chanted as the circumstances call.
I know of several monastic communities that will make a procession on the Feast of the Assumption and will chant the Litany of Loreto (in Latin) and end the procession at a statue of Mary (all the time remaining either walking or standing). If remaining in a church, these same communities will take the traditional posture of kneeling while reciting this litany. Once again this is only a traditional way of praying any of the litanies and the faithful are quite free to choose otherwise.
One last note: Please make sure to conclude the Litaniae Lauretanae with the appropriate closing prayer according to the liturgical season (Christmas, Easter, Advent, etc.).