How can I get notifications on my iPhone of Twitter hashtag mentions?
At first, I thought there would be a straight-forward, albeit hidden, solution. However, after doing some digging, it seems that something that should be a simple swipe or click to do is more complicated than that. This is going to be a little long, but hopefully will be incredibly helpful. I was going to post this earlier today, but did not get the chance to make this whole post until tonight.

Boxcar

From what I've seen, Boxcar.io is one of the best ways to get push notifications for tracking hashtag mentions. Both the iOS app and the web client allow a user to add a varying amount of sources to get information from. Twitter, email, RSS feeds, etc. can all be collected into Boxcar and keep track of all of your updates. Boxcar is not made specifically for what you are trying to do but the iOS app has the option of adding filters, where a group of sources can be viewed as one folder and have a key phrase that is picked out from all the sources to create a push alert.
What I did in Boxcar that worked was create an RSS feed using Queryfeed.net to generate an RSS feed for the term #hashtag. Queryfeed is extremely useful for this because Queryfeed can create an RSS feed for Twitter, Google+, and Facebook using specific search terms.

After adding all the sources that you may want (Twitter accounts, RSS feeds, etc.)  you can create a filter in the iOS app where you can select which sources you want in the feed, as well as other settings for updates.
There is a lot of customization that a Boxcar user can do, especially in tracking certain hashtags or trends. It is a little more work than some may want to put in and it is not problem free (in my experiments with Boxcar today there was definitely a lag in getting RSS updates at first, and the web client and iOS app are not exactly intuitive in their own unique ways) but it definitely does allow someone to track and receive alerts for specific hashtags.
IFTTT
There is another option that I won't delve into as much as I did Boxcar simply because in my experience, it is a little more finicky.
IFTTT is an app that operates in If-Then statements. An example they use on their about page is as follows: if I post a picture to Instagram, then save that photo to Dropbox. &quot;If&quot;s are called triggers (in this case, the mention of #hashtag on Twitter), and &quot;then&quot;s are called actions (getting an iOS push notification when a tweet is made using #hashtag).
Combined, triggers and actions make Recipes. A Recipe connects Channels (the channels you would want to use are Twitter and iOS Notifications). In essence, IFTTT links one application's actions to get a reaction from another.
While the user interface is a bit easier than Boxcar, IFTTT Recipes are one trigger, one action. You cannot, for example, have a feed like in Boxcar that searches for a specific term across multiple platforms, like finding the use of #hashtag in Facebook as well as Twitter. My (limited) personal experience with IFTTT is that occasionally Recipes don't want to work and I did not have the patience to figure out why. However, IFTTT is supported in Boxcar, so trying both and seeing which works best for you won't be much of a hassle.