In theory, yes. The portcullis was most often made of wood with some metal reinforcements, so was the gate itself. Application of some mixture that was both flammable and viscous could lead to the wooden elements being burned through or weakened enough to facilitate destruction with a kinetic force, such as the battering ram of even hand-held tools. But there are two main problems that anyone attempting such a feat would have faced. First, there were very few substances that could have been used to such effect short of Greek and Byzantine fire with the latter being of limited use as it required contact with water to ignite. Sure, there was pitch and tar widely available but ignition point of these substances are several times higher than that of gasoline and they generally need to be heated to 90-100 degrees Celsius to ignite. Any kind of oil was generally out of question, despite common use of 'burning oil' in fantasy. The flashpoint of animal and plant fats that were available in the Middle Ages are usually in the ballpark of 300 degrees Celsius (higher than the ignition point of dry wood), making them useless unless heated to the temperature similar to that of the melting point of lead.
The easiest way of application of such substance would have been shooting the heated container full of pitch from some kind of siege engine, with the clay jar being the most obvious as it can be easily heated to extremely high temperatures and will shatter on impact, releasing the burning payload. Such is the subject of the drawing made by Mariano di Iacopo, also known as 'Taccola', in his book 'De machinis' (on machines) published in 1449, where an incendiary barrel is shown being shot from a machine roughly resembling trebuchet [1].
Any attempt of applying such a substance by hand would be extremely risky and difficult, because the gates were usually separated from the approach by a moat and people trying to set the gate ablaze would become a target of keen interest from the defenders able to attack them through murder-holes or from battlements. Hand application of burning, viscous substance to a large structure is also not a easy trivial feat either. Heating the pitch at o a necessary temperature after the application is very hard without a modern blowtorch. But where is a will there is a way. Aforementioned Mariano di Iacopo envisioned a protective wheeled vehicle, not unlike the shielded battering ram with the barrel containing burning substance mounted on the end of the protruding beam [2]. Another illustration in the book shows the application of such a device precisely to damage or destroy the gates of the besieged city [3].
The incendiary mixtures could have been different. Treatise 'Liber ignium ad comburendos hostes' (Book of fire for burning enemies) attributed to a person known only as Mark the Greek and published in the late 13th or early 14th century (with a possibility of it being a translation of the Arabic original) mentions several dozens of recipes utilizing such ingredients as rosin, colophony, gum arabic, naphtha (these two suggest Arabic or Byzantine origins of the book), various plant oils, tallow, wax, pitch, dried bird droppings (good source of nitrates and phosphates), 'white, red and lustrous sulphur' (possibly galena, cinnabar and pyrite) and quicklime. Some kind of incendiary mixtures had to be known earlier though, as similar substances were used to attack wooden fortifications and buildings during siege of Montreuil-Bellay in 1147 as described in 'Historia Gaufredi ducis Normanorum' (History of Norman duke Geoffrey [Geoffrey Plantagenet]) by John of Marmoutier. Similar methods were also used to destroy the Crusaders' siege towers during siege of Acre in 1190 (as per Imad ad-Din, some sources claim that the towers were destroyed during the sally).
Of course, starting from the second half of 13th century one could have used the novelty that was the blackpowder, utilizing its explosive capacity. This would have worked well against the portcullis if the charges were set between the bars, less so against the gate door as the surface-mounted charges would have had little effect.
[1] Paolo Santini's copy of di Iacopo's 'De machinis' showing various incendiary devices. Online exhibit, Museo Gallileo.
https://brunelleschi.imss.fi.it/genscheda.asp?appl=LIR&xsl=paginamanoscritto&lingua=ENG&chiave=101235
[2] Mariano di Iacopo's drawing of a device for transporting incendiary kegs, Online exhibit, Museo Gallileo.
https://brunelleschi.imss.fi.it/ifoto.asp?foto=https://vitruvio.imss.fi.it/foto/ingrin/ingrin-53MAC10rs.jpg
[3] Another illustration from 'De Machinis' by Taccola, showing the application of the 'fire cart'.
