I'd like to provide an answer to my question based on the material that's been provided, as well as my own investigation into this matter. Thank you Mauro ALLEGRANZA and virmaior for the very helpful resources.
Joseph Priestley has had a very significant influence on modern philosophy. Most of his philosophical influence comes from his redaction of David Hartley's Observations of Man, and from two essays; Disquisitions relating to Matter and Spirit and The Doctrine of Philosophical Necessity Illustrated. Priestley's philosophy was a mixture of determinism, materialism, and theism into what he called "philosophical necessity". He denied mind-body duality (believing it to be a Platonic corruption of Christianity), freewill, the trinity, and eternal hell.
Jeremy Bentham, who is considered the founder of Utilitarianism, wrote in Commonplace Work in Books:
"Priestley was the first (unless it was Beccaria) who taught my lips to pronounce this sacred truth:- That the greatest happiness of the greatest number is the foundation of morals and legislation."
Although Beccaria did say something similar, Bentham gives the credit to Priestley for his entire philosophy of utilitarianism! In Priestley's The First Principles of Government and the Nature of Political, Civil and Religious Liberty, he says:
"The good and happiness of the members, that is the majority of the members of the state, is the great standard by which every thing relating to that state must finally be determined."
Also, it should be mentioned that Francis Hutcheson wrote in 1725:
"that Action is best, which procures the greatest Happiness for the greatest numbers; and that, worst, which, in like manner, occasions Misery.”
However, there is no evidence that Bentham knew of Hutcheson.
Another Utilitarian, John Stuart Mill, praises Priestley for consistently promoting David Hartley's book Observations on Man. Priestley had written a redaction of Observations, excluding much of Hartley's technical material and adding his own interpretation. Mill said in his Essay on Ethics, Religion, and Society:
But for the accident of their [Hartley's Doctrine] being taken up by Priestley, who transmitted them as a kind of heirloom to his Unitarian followers, the name of Hartley might have perished, or survived only as that of a visionary physician, the author of an exploded physiological hypothesis.
Mill quoted Priestley frequently, and in a reply to William James Ward in Essays on the Philosophy of Theism, Ward quotes Mill as saying:
The answer made by Priestley, in his examination of Reid, seems to me sufficient, that though we have had no experience of what is future, we have had abundant experience of what was future.
These are a few quotes I've been able to dig up in the past few days.
Priestley was also a member of the Lunar Society of Birmingham, a dinner club of Enlightened intellectuals and natural philosophers. Wikipedia says:
In late 1780 the nature of the society was to change again with the move to Birmingham of Joseph Priestley. Priestley had been closely associated with the group's activities for over a decade and was a strong advocate of the benefits of scientific societies. Shortly after his arrival Lunar meetings moved from Sunday afternoons to Mondays to accommodate Priestley's duties as a clergyman, while the society's dependence on Matthew Boulton was lessened by holding meetings at other members' houses in addition to Soho House. The result was to be the society's most productive era.
Prior to Priestley moving to Birmingham, he was already greatly involved with the activities of the Lunar Society, meeting weekly with many great minds, including Erasmus Darwin (the grandfather of Charles Darwin). Wikipedia says:
The Lunar Circle also attracted more distant involvement. Joseph Priestley, then living in Leeds and a close friend of John Mitchell, became associated with the Society in 1767 when Darwin and Wedgwood became involved with his work on electricity. In the same year James Watt visited Birmingham on the recommendation of his business patron John Roebuck, being shown around the Soho Manufactory by Small and Darwin in Boulton's absence. Although neither Priestley nor Watt were to move to Birmingham for several years, both were to be in constant communication with the Birmingham members and central to the circle's activities from 1767.
The group eventually ended after what has been coined the Priestley Riots of 1791, when Priestley's home and church were burned to the ground.
Finally, for everyone that follows Immanuel Kant, this is what Kant said about Priestley in Critique of Pure Reason:
If we could ask that dispassionate philosopher, David Hume, who seemed made to maintain the most perfect equilibrium of judgment, what induced him to undermine by carefully elaborated arguments the persuasion, so useful and so full of comfort for mankind, as that reason is sufficient to assert and to form a definite concept of a Supreme Being, he would answer, Nothing but a wish to advance reason in self-knowledge, and at the same time a certain feeling of indignation at the violence which people wish to inflict on reason by boasting of her powers, and yet at the same time preventing her from openly confessing her weakness of which she has become conscious by her own self-examination. If, on the contrary, you were to ask Priestley, who was guided by the principles of the empirical use of reason only and opposed to all transcendental speculation, what could have induced him to pull down two such pillars of religion as the freedom and immortality of our soul (for the hope of a future life is with him an expectation only of the miracle of a resuscitation), he, who was himself so pious and zealous a teacher of religion, could answer nothing but that he was concerned for reason, which must suffer if certain subjects are withdrawn from the laws of material nature, the only laws which we can accurately know and fix. It [p. 746] would be most unjust to decry the latter, who was able to combine his paradoxical assertions with the interests of religion, and to inflict pain on a well-intentioned man, simply because he could not find his way, the moment he strayed away from the field of natural science. And the same favour must be extended to the equally well-intentioned, and in his moral character quite blameless, Hume, who could not and would not leave his abstract speculations, because he was rightly convinced that their object lies entirely outside the limits of natural science, and within the sphere of pure ideas.
That's all I have for now. Joseph Priestley was a fascinating man, and for any philosopher that believes reason still holds an important place in philosophy, I highly recommend his work.