In 5e, are alcoholism and drug addiction "curable" diseases? Could they be cured by Lesser Restoration and other spells/items that "end diseases"? Could symptoms of withdrawal be cured/relieved by healing spells/items?
As with any topic that intersects with real world behaviors and beliefs, you need to be both careful and considerate in using your game to explore topics of alcoholism or drug addiction. For all you know, one or more of your players has either been impacted or is currently being impacted by these issues, either personally or through a close friend, relative, or loved one.
Most likely because of the sensitivity of such issues, and being careful not to trivialize or 'gamify' the issues involved, the RAW content of the game does not have much to say about either.
The DMG, for example, has exactly two references to alcohol (if my word search is to be trusted). One is as a solvent (not the kind of alcohol one would drink) on p. 209, and the other is as one of the methods that a villain might use to achieve their nefarious ends (p. 95, Villain's Methods Table, result 19 (Vice), 2 (Drugs or Alcohol)).
My suggestion would be that you treat alcoholism and drug addiction as 'the DM describes the environment', in a way that serves your plot and the story you are telling rather than relying on the game to provide a rules structure for it.
RAW, probably an indefinite madness
However, if you are seriously interested in a RAW approach, alcoholism and drug addiction should not be treated as diseases. As Groody the Hobgoblin explains in detail in their answer, multiple references point to the rules considering diseases to be what we would more specifically call infectious diseases. Rather than disease, alcoholism and drug addiction fit the game better as a form of 'madness' (and see pages 258 - 260). Looking through the descriptive examples, I would suggest them to be forms of 'indefinite madness':
A character afflicted with indefinite madness gains a new character flaw from the Indefinite Madness table that lasts until cured.
The flaw "Being drunk keeps me sane" seems an appropriate model, when tailored to the particular story of the individual character, of the form 'I believe that using [X], which has negative consequences [Y], is necessary because otherwise I would have negative consequences [Z].'
As an indefinite madness, these would not be curable by lesser restoration, although a greater restoration would work.
The symptoms of withdrawal, literally dis-eases, could be (temporarily) relieved by lesser restoration and other effects that cure disease.
For some discussion of trying to overcome an addiction through roleplay, rather than magical means, see this Q&A. It is locked as no longer on-topic for the site but of historical significance, and includes a thoughtful answer by Frank Metzner.
They most likely are character flaws
There is no definition of what counts as a disease in 5e, but from the text in the DMG and from the sample diseases, it is clear that the game thinks more of infectious diseases than of inborn defects, addictions or mental disorders. It talks about outbreaks. The DMG says on page 256:
A disease that does more than infect a few party members is primarily a plot device.
Which suggests that diseases are infectious. Furthermore, diseases have incubation times, which is not something alcoholism or other addictions have, instead it is associated with infectious diseases.
You most likely consider such addiction diseases as indefinite madness under the game mechanics instead. The DMG on p 261 has this on the indefinite madness table:
Being drunk keeps me sane.
Indefinite madness mechanically is a character flaw. According to p. 260 DMG you can only cure it with greater restoration, or an even more powerful healing spell.
A greater restoration spell or more powerful magic is required to rid a character of indefinite madness.
As a recovering addict, I once scoffed at any statement that referenced addiction as a "disease". It took nearly 2 decades in active addiction, a 2-year strict, mandatory rehabilitation program, and further medical support for me to learn addiction rewires your brain - programs neurons - and I finally understood the reason it could be considered a disease.
That doesn't answer your question, though. Personally, just in my opinion, I can't honestly tell someone I believe addiction is "curable". For lack of a better comparison, it's like HIV. It's treatable, though any true, real addict is never fully cured.
Like the lyrics to Hotel California...
You can check out any time you like... but you can NEVER leave...
The addict inside of us will never forget what it was like to get high. Our addictive personalities will never be fully free from the temptation or desire to go back to that life. Hence it is my opinion that it is absolutely NOT curable, but it can be treated when that person truly wants help and treatment.
And with this type of treatment, much like HIV, you can go the rest of your life without developing AIDS - i.e. touching another substance or getting high.
Neither "alcoholism" or "drug addiction" are defined in the rules, so this is firmly in the rulings and house-rules territory, and can't be answered purely based on the books.
Addiction is a complex thing, which includes both physiological and psychological addiction, and several types of both. On the psychological side, there's the force of habit, and then there's the desire to get the "rush" or "high", and then there can be the desire to "dull the pain" of something. On physiological side, there are the "hang-over" type poisoning symptoms, where the break-up products of the substance cause poisoning type symptoms, and then there are more persistent changes in the body physiochemistry to adapt to constant presence of the substance in the body, and then there is actual damage caused by the substance, both taking it in and processing it.
Any "reduced maximum HP" type damage would need Greater Restoration. This might be something like a damaged liver. I could also see Regeneration allowing to re-grow damaged tissue (both internal and external). Cure Wounds might not do much here, it's more for immediate damage which could be healed by taking a short rest and using hit dice.
If there are any actual conditions caused either by the substance or as withdrawal symptoms, Lesser Restoration would remove those for the moment, but they might come back, because the root cause has not been removed. Also, I'd qualify the issues caused by going "cold turkey" and suddenly changing body chemistry after heavy/prolonged substance use as a disease, and allow Lesser Restoration to cure that (possibly avoiding death, if you want to go gritty).
"Hangover" type issues are basically poisoning, so Protection from Poison could be used. Lesser Restoration is also pretty traditional as general "Aspirin" type remedy to hang-over, or at least has been in the games I've been in.
Then to the hard part... Psychological addiction:
Modify Memory might be useful for breaking psychological addiction, such as by giving memory of such a horrible hang-over, that even thinking of drinking would make a person feel sick.
Persistent use of Calm emotions might also be utilized to help a recovering individual to overcome their psychological desire to re-start using the substance. Suggestion is an obvious short term remedy of this type as well, to temporarily prevent the recovering person from wanting to re-start.
Geas can also be used to force a person to not use a substance for 30 days (or forever with 9th level spell slot), though it can be quite fatal to normal people breaking it, with its 5d10 damage.
For actual madness caused by prolonged substance abuse, that is well covered in the rules actually. The short of it is, Calm emotions will suppress madness, Lesser Restoration can actually cure normal madness, Greater Restoration should be able to cure any madness.
But for a person to actually not want to use a substance any more, ever, I don't think there is any magic which can do that permanently (except Wish and Divine Intervention of course).
In the end, since we are in the house rule territory, it's up to the DM and their goals for the campaign. If the addiction is not central to the plot or the arc of the afflicted character, Lesser Restoration should be good enough. If it's supposed to be a puzzle or challenge to overcome, then I'd expect something like organ changes caused by the mutation, requiring daily Lesser Restoration until the party is able to do both Greater Restoration and Regeneration to restore the body properly, followed by baby-sitting or above mentioned spells to break the psychological addiction.