I wonder if burning incense at home is allowed in the Syriac Tradition.
Our tradition (Syriac as we have received it in India) concerning incense is basically along the following lines.
In any liturgical service, only a bishop or a priest may bless and place incense in the censer because the offering of incense is considered a sacrificial act, and sacrifice is proper to the priesthood*. If a bishop is present, whether or not he "leads" the service, he will offer the incense unless he specifically allows any priest(s) present to do so. The only exception to this is in the case of a full deacon: if he is leading a service because there is no priest, he can offer incense but without blessing it.
While it is always the prerogative of the bishop/priest to cense (swinging the censer), this can and often is delegated to deacons. Obviously, the best practice is to give this to a full deacon, but this is often "outsourced" to one of those in the minor orders. They bring the censer to the priest, receive the incense, and are given a blessing, after which they cense the altar, the gospel, and the church and the people in order. In the case of a full deacon offering incense (as I described above), he must do the censing himself because he cannot give a blessing in order to delegate the task to someone else.
Incense is not burned at home in the context of prayer unless a priest or a full deacon is present. Perhaps some might burn church incense over coals at home for the fragrance, but I think in India this is not done. If any incense is burned at home for its fragrance, it is usually in the form of incense sticks. The incense we use liturgically is pure frankincense, we don't use the perfumed stuff common among EO and some of the OO. If we want to add fragrance to the liturgical incense, what I've seen done is the mixing of frankincense with crushed incense sticks: these are offered together in the censer.
*I forgot to add this: when a priest is ordained in our tradition, after the bishop vests him, he presents the new priest with a censer and allows him to offer incense on his own for the first time as a priest. In some traditions, the bishop will step aside and let the newly ordained complete the Liturgy, but in India it doesn't always happen that way.