I’m wondering what type of lumber/spacing of joists would be able to handle a hot tub on an exterior deck? I’d like a larger, permanent fixture which can have at least 6-8 people, but I’d also settle for one of those inflatable hot tubs that will hold up to 4 people and are only about 250 gallons of water.
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I would recommend hiring a licensed Structural Engineer. It doesn’t necessarily mean that is going to be too expensive. The reason for that is that many factors play a role like soil types, adjacent structures, deck height, and most importantly, the hot tub weight with water (not too good in seismic zones). I have seen quite different requirements from Local Building Departments; everything from regular decks supported with independent footings, all the way up to full decks with grade beams (similar to a regular house foundation) with shear and fire rated walls.
From the liability perspective, you will save more money in the long run if you hire a Structural Engineer for that purpose.
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Excellent recommendations to hire a structural engineer to analyze the existing framing system of your deck and recommend modifications if needed.
Issue’s of concern:
1-Load of hot tub with water & people.
2-Methods of securing the deck agaist the home.
3-Proper footings under deck supports.
4-Profide you a sketch / drawings to obtain building permit.
Great question! Our favorite topic at DDT Home Transformations is decks, especially Trex. 
The answer to your question is…‘maybe yes’ or ‘possibly no’. Without being too redundant (because the other answers address this), your “foundation” must support the weight of the tub.
For instance, my toddler likes to get on my shoulders in big crowds, so she can see. I can support the mass of my daughter. If I wanted to get on top of someone’s shoulders to see, would I choose my daughter’s shoulders to support me? In other words, it doesn’t matter how much weight you have. It matters if you build for it, so the substructure can support it.
Remember that water weighs 8.35 lbs. per gallon.
A typical hot tub holds 400-600 gallons of water.
That’s 3,340 - 5,010 lbs. of water.
The hot tub itself will weight between 500-700 lbs.
So, the anticipated total weight is between 4,000 - 6,000 lbs.
And, don’t forget about the 4-6 people at another 900 lbs.
Where the hot tub will be placed will require additional substructure material and thicker material. You will need a structural engineer or architect to provide accurate structural information.
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When planning a new deck this should be considered in your drawing by the Engineer who will just add the proper support for that item.
If renovating an existing deck I refer back to the need for an Engineer…
Engineer " GOOD"
In most cases if you want to install a hot tub on your deck, the deck will need to be engineered for the additional weight.