How do I convince my professor to allow me to go to a conference?
Tight question and hard to answer without knowing the details. It may be that your advisor is right in that your results are not good enough to publish and that other colleagues you ask are just polite and it may be that your advisor is wrong…

From the perspective of your supervisor it looks like this: Sending you to a conference will cost money from his budget or grants and it can contribute to his reputation if good results are presented from his work group and it could harm is reputation if bad/shaky/unfinished work is presented. On top of that, your advisor has genuine interest to promote his students appropriately. However, many supervisors do not send students to conferences without own contributions - maybe a rationale behind this is that you have to earn this privilege.

Let me comment a bit on the things you wrote. You say your arguments have been


  a) discuss my work with other people and possibly get some new ideas and 
  b) I want to get a postdoc later and need to do some networking, if possible, at an international level.


I can relate a little bit to the response of your supervisor. Conferences are great for sharing ideas and for networking but it actually easy to go to a conference and fail in  both respects. Often there are tons of people, many knowing each other, other actively networking and for somebody new (especially if a bit shy) it is not straightforward to get involved. In fact, it is harder to network at conferences that to share ideas within your workgroup. Also, a talk at a conference is a great way to start networking and without a contribution, networking gets even harder.

You got the advice to try more local conferences in a comment and responded


  There were some local conferences, but only the people from our (small) community presented. I mean that if you go and see only your colleagues' talks, it will not really help you in networking...


In view of the above, this sounds like a strange response. You say you don't think networking on the local level is beneficial but only international networking is helpful? That doesn't make sense. Start small, practice networking with locals and then go for more. Also "local" does not necessarily mean you workgroup or university, but could mean "within your region or country". Actually, networking within the own region and community is very important (and a bit different than international networking).

In conclusion, the conflict between you and your supervisor seems to be, that he thinks that your contributions are not good enough for the conferences you proposed and you disagree on this.

So here is a suggestion: You should get qualified external feedback on your work, probably even anonymous:


Submission to a journal. Does not seem like a good idea. The feedback will take a long time, higher chances of rejection (even with mildly positive feedback). On the positive side, the feedback can be quite detailed.
Submission to a mid prestige conference with peer review. On the positive side, there is a fixed time frame for the feedback. Depending on your field, the feedback may be very brief.
Find a double blind refereed venue (so a bad feedback would not harm your supervisor reputation). Pro and cons for conferences and journals are the same as above.


Then there was this piece:


  one of the opponents said that it could be beneficial for me to go to another (specific) lab for a couple of weeks


To go from here, contact that very person and if you can manage to get an invitation from that lab, your supervisor should be happy to let you go (if the person is not from this lab, you may ask him to introduce you to the people from that lab). Even more so, if the lab could even contribute to the costs of the visit.