I think it's more partisan intra-party, where Conservatives are more partisan outside of their party rank-and-file. The left isn't quite as homogenous and ideologically pure as the Right - I think that's why we have a Tea Party and the Conservative, establishment, "country club" republican party as two quite separate, distinct political entities. There's isn't a liberal equivalent of the Tea Party shooting off from the "Clinton democrats" just yet, but with the advent of Bernie Sanders, we could very well see that, or he could turn the entire party even more left, which I feel is more likely.
Here's something I just read on Wikipedia:
As presidents, Bill Clinton and Barack Obama both reflected the priorities of the second New Democrat coalition, uniting donors from Wall Street, Hollywood, and Silicon Valley with a "new majority" coalition of racial minorities, immigrants, liberal women, and young voters. Because Democratic voters are disproportionately poor, this has produced a Democratic Party that, in economic terms, is an hourglass coalition of the top and the bottom. Economic populism frightens the party's billionaire donors, while social populism, which has often been associated with white working-class xenophobia, racism and religiosity, frightens blacks, Latinos, immigrants and white social liberals. The result is what Mike Konczal and others have called "pity-charity" liberalism — a kind of liberalism that appeals to the sympathy of the rich for the poor, rather than appealing, as the New Deal did, to solidarity among the middling majority.
Would the party turn left? I don't think so, the Democratic party could probably win elections anyways. Blacks will overwhelmingly vote Democrat, Latinos vote also vote quite strongly for Democrats and will probably continue to do so as long as Republicans don't budge on immigration reform, Asian Americans tend to be more affluent and can be appealed with low taxes, but they're a small part of the voting bloc and apparently have low turnouts so electorally they're not very important. The two parties will compete for white voters who make up about 75% of the electorate (
source), but the Democrats have a distinct demographic advantage that won't go away any time soon.
Based on what I understand, the Democratic party shifted rightwards towards the centre to appeal more to white voters who wanted fiscal conservatism starting with Clinton being the first president to embody this movement and continuing with Obama, and likely Hillary Clinton if she gets the nomination. If the political culture of the US doesn't change, then I think the Democratic party would want to hold on to those white voters it's captured. I think there's no reason to change because the minority bloc is voting for the Democrats anyways, and they need as many white voters as they can get.
I'm not very informed on how election turnouts get into the mix as well as the intricacies of the electoral college, so everything I've said might be distorted.
On a "Tea Party" for the democrats, I'm not too optimistic about that either. The Tea Party had clear political initiatives that it could oppose: the bailout and health care reform. Fighting income inequality and returning democracy to the people could be just as sexy, but it's just business as usual and I don't think there's a big event or happening that people can point to. Furthermore, the Tea Party movement fights for lower taxes and reduced government intervention, which can appeal greatly to business elites and attract political donations as well as media attention from them. I'm sure there are anti-corporate Tea Partiers as well but as far as corporate America is concerned, that's mere rhetoric as long as they remain like-minded on economic policy. The kind of movement that Sanders is inspiring would get zero support from the elite. I suppose this is the kind of scenario where the trade union movement could come into play but, in my understanding, it is weak in the United States.
I think the only solution is making income inequality and democratic reforms really really sexy. That'd give the Democratic party a good reason to move to the left, and that'd also compensate for the lack of elite support for an influential progressive movement. Trouble is that one does not simply "make" issues prominent.
I've probably left a thing or two out - I don't live in the United States, so I don't have the same on-the-ground experience that allows you to better assess what's relevant and what isn't. What are your reasons for believing that Sanders could turn the Democrats more to the left? Believe me, I wish that would both be true and not cost the Democratic party the White House and Congress.