Does Hinduism permit kings to act tyrannical? As in allowing kings to exploit the people, loot them, kill, take women, etc?
No, Hinduism forbids kings from acting tyrannical.
Some verses from the Manusmriti:
The king, who, without affording protection, takes tributes, taxes, duties, presents and fines, would immediately sink into hell.—(8.307)
He who affords no protection and devours the people, grabbing his tribute of the sixth part of the produce,—him they declare to be the imbiber of the filth of the whole people.—(8.308)
He who heeds not the bounds of morality, who is a disbeliever, who is extortionate, who does not afford protection, and is grabbing,—such a king one should regard as doomed to perdition.—(8.309)
He who, on being abused by men in distress, forgives, becomes exalted to heaven, by that act; while he who, through kingly pride, does not forgive, goes, by that act, to hell.—(8.313)
So a king who is unrighteous and oppresses his people goes to hell.
Also, just like the US 2nd amendment, one can take up arms to defend himself and others, even against a tyrannical king, according to the Manusmriti:
Twice-born persons shall carry arms: When religion is interfered with, when there is confusion among the twice-born castes caused by the exigencies of time,—(348) in his own defence, in cases of hindrance of sacrificial fees, in the case of outrages upon Brāhmaṇas and women,—if one strikes in the cause of right, he incurs no sin.—(8.348-8.349)
And Medhatithi's commentary for that verse:
Another interpretation possible is that—“when religion is interfered with, when there is confusion caused by exigencies of time, i.e., when things have become unsettled on the death of a king—one may take up arms; but at other tiroes the necessary protection would be afforded by the king himself.”
But in reality the king cannot spread out his hands and reach every individual person in the kingdom. There are some desperados who attack even the boldest, and the most trusted officers of the king; but they fear persons carrying arms.
For these reasons it is right that one should carry arms at all times.
‘Caused by the exigencies of time,’—such as the death of the king, and such other calamities. On all these occasions one shall carry arms fur the protection of his properly and family.
Others hold that on the occasions stated, arms may be carried for the sake of other people also;—says Gautama (21.19)—‘Also when some one is striking a weaker man, if he is able to rescue him.’
Additional commentary called Madanapārijāta:
if there is interference with the sacred duties due either to the tendencies of the king or to the tendency of the times
And commentary called Mitākṣarā:
when, on the waning of royal authority due to foreign invasion, one has to take care of himself
No. Kings who act tyrannically will be destroyed.
Vamadeva continued, ‘When the king, who is powerful, acts unrighteously towards the weak, they who take their birth in his race imitate the same conduct. Others, again, imitate that wretch who sets sin agoing. Such imitation of the man ungoverned by restraints soon begins destruction upon the kingdom. The conduct of a king who is observant of his proper duties, is accepted by men in general as a model for imitation. The conduct, however, of a king who falls away from his duties, is not tolerated by his very kinsfolk. That rash king who, disregarding the injunctions laid down in the scriptures, acts with high-handedness in his kingdom, very soon meets with destruction. That Kshatriya who does not follow the conduct observed from days of old by other Kshatriyas conquered or unconquered, is said to fall away from Kshatriya duties.’
Mahabharata, Santi Parva, Section XCIII
When the Rishis were moreover insulted by the pretentious scholarship of Vena, and when they found that their noble prayer for the welfare of the world as a whole was rejected, their anger was aroused, and they said: Let him be destroyed, let him perish. He is by nature a perverted monster. To let him live is to allow the worlds to be reduced to ashes. An evil person like him deserves not to sit on the throne.
Srimad Bhagavata Purana IV.14.30-32
The King derives his highest good by protecting his people. A King who protects his people well, will derive one-sixth of the merits of his subjects in the life hereafter. But a King who collects taxes from people without administering their affairs properly, will lose all the merits to his credit and will inherit the sins of his people to boot.
Srimad Bhagavata Purana iV.20.14